1 00:00:05,000 --> 00:00:07,700 Okay, what's the first thing we always do? 2 00:00:09,720 --> 00:00:10,620 Direct substitution. 3 00:00:14,820 --> 00:00:26,400 Okay, zero squared times sine of, what does one over zero go to? 4 00:00:30,400 --> 00:00:37,923 Any number in terms of limits over zero goes to either positive or negative 5 00:00:37,923 --> 00:00:38,673 infinity. 6 00:00:41,040 --> 00:00:43,880 We don't know until we do it from the left or from the right. 7 00:00:47,200 --> 00:00:48,100 Let's just call it infinity. 8 00:00:50,640 --> 00:00:55,520 And what does sine of infinity or sine of a very large number do? 9 00:00:56,540 --> 00:00:57,440 Go to. 10 00:00:58,140 --> 00:00:59,040 What's that? 11 00:01:02,220 --> 00:01:08,260 Let's, uh, so this goes to one of these. 12 00:01:12,440 --> 00:01:14,720 What does sine do as x gets larger and larger and larger? 13 00:01:17,460 --> 00:01:19,200 Think of the sine function, what does it do? 14 00:01:20,160 --> 00:01:21,440 It just keeps going, oscillating. 15 00:01:22,760 --> 00:01:23,960 It does not approach a number. 16 00:01:25,260 --> 00:01:26,160 Okay? 17 00:01:26,560 --> 00:01:29,740 Since sine does not approach a number, this does not exist. 18 00:01:34,420 --> 00:01:37,820 So zero times does not exist, I have no idea what that means. 19 00:01:39,360 --> 00:01:43,635 Okay, so the function value definitely doesn't exist zero, there's a hole there, or 20 00:01:43,635 --> 00:01:44,385 there's 21 00:01:43,940 --> 00:01:46,834 There may be a hole or some other behavior at zero; direct substitution has not 22 00:01:46,834 --> 00:01:47,584 settled it. 23 00:01:49,220 --> 00:01:51,640 Okay, so anyways, direct substitution does not work. 24 00:01:53,260 --> 00:01:54,160 So, 25 00:01:56,180 --> 00:01:57,520 we need another trick. 26 00:01:59,940 --> 00:02:03,220 It's one of our limit laws. Why don't you take out that limit laws. 27 00:02:03,220 --> 00:02:05,900 Okay, the very last one is called the squeeze theorem. 28 00:02:33,160 --> 00:02:34,560 And 29 00:02:34,560 --> 00:02:37,740 has two hypotheses. 30 00:02:38,320 --> 00:02:40,260 Let me look at what a hypothesis is in math. 31 00:02:42,200 --> 00:02:46,060 So, most theorems in math are an if 32 00:02:47,080 --> 00:02:50,240 something is true, then 33 00:02:50,240 --> 00:02:55,160 something is true. So all the things that must be true are called the hypotheses. 34 00:02:59,780 --> 00:03:01,240 You don't have to write that down. 35 00:03:01,920 --> 00:03:05,892 Okay, so this has two hypotheses, two things that have to be true for the conclusion 36 00:03:05,892 --> 00:03:06,642 to 37 00:03:06,140 --> 00:03:07,040 be true. 38 00:03:08,000 --> 00:03:16,500 Okay, so first off, we have to have three functions. 39 00:03:18,400 --> 00:03:21,909 The middle function must be always greater than some function and always less than 40 00:03:21,909 --> 00:03:22,659 some 41 00:03:22,160 --> 00:03:23,060 other function. 42 00:03:24,440 --> 00:03:30,314 and the limit as x approaches some number of this one and this one must equal the 43 00:03:30,314 --> 00:03:31,064 same 44 00:03:30,660 --> 00:03:31,560 number. 45 00:03:33,580 --> 00:03:41,920 So to use the squeeze theorem, we have to take our function and make sure it's bound 46 00:03:41,920 --> 00:03:47,480 just oscillates back and forth, they're always bound between two other functions. 47 00:03:49,540 --> 00:03:53,926 What's the largest, no matter what x is, what's the largest sine of some angle will 48 00:03:53,926 --> 00:03:54,676 be? 49 00:03:55,440 --> 00:03:56,340 1. 50 00:03:57,100 --> 00:03:58,380 Right, so here's the graph. 51 00:03:59,400 --> 00:04:00,300 Is this right? 52 00:04:00,960 --> 00:04:06,620 The maximum value here is at 1, and the minimum value is negative 1. 53 00:04:06,620 --> 00:04:26,404 So, this function, which is y equals 1 and y equals negative 1, it's bound between 54 00:04:26,404 --> 00:04:27,640 those. 55 00:04:27,640 --> 00:04:35,720 So we can always write this inequality. 56 00:04:37,300 --> 00:04:42,367 This is always less than or equal to one and always greater than or equal to 57 00:04:42,367 --> 00:04:43,117 negative one. 58 00:04:46,300 --> 00:04:50,060 The expression is at most one and at least negative one. 59 00:04:50,980 --> 00:04:53,780 You can have all kinds of functions inside the sine function. 60 00:04:53,780 --> 00:04:58,721 When we have sine of something we always know it's bound between negative one and 61 00:04:58,721 --> 00:04:59,471 positive one. 62 00:04:59,380 --> 00:05:05,220 And the only thing we're missing is this guy right here, okay, so we take this 63 00:05:19,600 --> 00:05:25,020 And we multiply all three sides by x squared. 64 00:05:33,460 --> 00:05:37,120 We multiply an inequality by a function or a number. 65 00:05:37,480 --> 00:05:39,140 When do we reverse the sign? 66 00:05:42,140 --> 00:05:43,760 When do we reverse the inequality? 67 00:05:44,520 --> 00:05:45,420 If it's negative. 68 00:05:45,360 --> 00:05:47,180 will x squared ever be negative? 69 00:05:48,280 --> 00:05:49,760 No, so we don't need to do that. 70 00:05:50,860 --> 00:05:53,960 If we did, if we just multiply by x, for example, 71 00:05:54,900 --> 00:05:57,760 this will still work out, but you have to do separate cases, 72 00:05:58,140 --> 00:05:59,260 but let's not worry about that. 73 00:05:59,780 --> 00:06:01,320 Okay, so now we got this inequality. 74 00:06:01,320 --> 00:06:04,800 And now this is that middle function here. 75 00:06:07,800 --> 00:06:13,360 And now we simply take the limit as x approaches zero of all three sides. 76 00:06:15,220 --> 00:06:18,420 We go the limit as x approaches zero. 77 00:06:18,420 --> 00:06:25,160 The limit as x approaches zero. The limit as x approaches zero. 78 00:06:25,160 --> 00:06:28,480 And what is the limit as x approaches zero of negative x squared? 79 00:06:30,280 --> 00:06:31,180 Zero. 80 00:06:31,860 --> 00:06:32,820 Just direct substitution. 81 00:06:33,140 --> 00:06:34,320 It's a continuous polynomial. 82 00:06:35,200 --> 00:06:36,100 That's zero. 83 00:06:37,680 --> 00:06:39,840 The limit as x approaches zero of x squared? 84 00:06:41,900 --> 00:06:43,320 Zero direct substitution. 85 00:06:48,400 --> 00:06:54,760 So this limit must be less than or equal to zero and greater than or equal to zero. 86 00:06:54,960 --> 00:06:56,240 So what must this limit equal? 87 00:06:56,960 --> 00:06:57,860 Zero. 88 00:06:58,600 --> 00:07:02,020 So we often write by the squeeze theorem. 89 00:07:08,220 --> 00:07:17,400 The limit as x approaches 0 of x squared sine of 1 over x equals 0. 90 00:07:26,300 --> 00:07:27,940 And when do we use the squeeze theorem? 91 00:07:29,720 --> 00:07:36,080 only special cases typically involving a sine or a cosine. 92 00:07:38,440 --> 00:07:45,260 Because we can always bound sine between one and negative one, same with cosine. 93 00:07:47,060 --> 00:07:47,960 Does that make sense? 94 00:07:48,660 --> 00:07:49,560 You just have to... 95 00:07:50,240 --> 00:07:51,360 How do I know to do that? 96 00:07:51,420 --> 00:07:55,380 Well, if everything else doesn't work, try it. 97 00:07:55,380 --> 00:08:02,380 And typically it involves a trig function. Let me graph these functions. 98 00:08:02,380 --> 00:08:03,280 x squared. 99 00:08:13,880 --> 00:08:16,860 Oh, geez, what did I do? To the power of sine of x? 100 00:08:16,860 --> 00:08:19,240 That's kind of cool. That's not the function. 101 00:08:23,920 --> 00:08:25,100 There it is. 102 00:08:27,420 --> 00:08:31,862 The function x squared times sine of one over x is always bounded between those two 103 00:08:31,862 --> 00:08:32,612 functions. 104 00:08:33,980 --> 00:08:36,100 If I zoom out. 105 00:08:41,060 --> 00:08:48,040 If you zoom in you see green functions are functions. 106 00:08:48,700 --> 00:08:51,640 You get squeezed in from both sides. 107 00:08:51,880 --> 00:08:53,040 That's why we call this squeeze theorem. 108 00:08:53,040 --> 00:08:55,700 sometimes called referred to as the sandwich theorem. 109 00:08:56,660 --> 00:08:59,540 That function's a sandwich between two functions. 110 00:09:03,880 --> 00:09:05,060 But there it is. 111 00:09:07,980 --> 00:09:10,040 Okay, so that's the squeeze theorem. 112 00:09:10,040 --> 00:09:14,120 Is this function continuous or over the numbers? 113 00:09:14,120 --> 00:09:27,989 No. No. That's discontinuous x equals four. Is it continuous x equals seven? Yes. 114 00:09:27,989 --> 00:09:28,980 How 115 00:09:28,980 --> 00:09:33,000 last year or the year before, how do we define what it meant to be continuous? 116 00:09:40,280 --> 00:09:41,900 There's no break in the function. 117 00:09:43,700 --> 00:09:47,787 Some of your teachers might have said if you can write the whole function without 118 00:09:47,787 --> 00:09:48,537 lifting 119 00:09:48,060 --> 00:09:49,800 up your pencil, it's continuous. 120 00:09:51,520 --> 00:09:52,800 That's not really a good definition. 121 00:09:53,760 --> 00:09:55,320 So we have a mathematical definition. 122 00:09:56,300 --> 00:10:01,160 And they didn't define it for you before because you didn't know what a limit was. 123 00:10:03,380 --> 00:10:07,800 So, the first thing we are going to do is define what it means to be continuous at a 124 00:10:07,800 --> 00:10:08,700 number. 125 00:10:10,680 --> 00:10:15,300 And here is the definition. 126 00:10:17,160 --> 00:10:18,340 Then we will all explain. 127 00:10:18,340 --> 00:10:19,940 I want you to write it down on the off-sides. 128 00:10:19,940 --> 00:10:25,280 Okay, so the definition, a function is continuous 129 00:10:25,280 --> 00:10:28,580 at some number, not on a set of numbers. 130 00:10:29,280 --> 00:10:34,220 At a single point, if, has to be defined, 131 00:10:35,400 --> 00:10:38,800 the limit must exist as x approaches that number. 132 00:10:39,840 --> 00:10:41,280 Really is all we care about. 133 00:10:43,900 --> 00:10:46,400 The limit as x approaches that number 134 00:10:46,400 --> 00:10:47,880 must equal the function value. 135 00:10:47,880 --> 00:10:53,620 When the limit equals the function value, the function is continuous at that value. 136 00:10:57,100 --> 00:10:59,260 It is continuous at that value. 137 00:11:04,300 --> 00:11:06,060 So here 138 00:11:06,060 --> 00:11:09,780 is continuous at 7 because the limit 139 00:11:09,780 --> 00:11:15,440 as X approaches 7 equals F of 7. 140 00:11:19,400 --> 00:11:20,300 It's 141 00:11:19,700 --> 00:11:23,040 discontinuous at 4 because the limit 142 00:11:23,040 --> 00:11:27,800 as X approaches 4 does not exist. 143 00:11:27,800 --> 00:11:29,800 So it definitely cannot be continuous. 144 00:11:35,860 --> 00:11:40,000 And then the next thing is it's continuous on an interval. 145 00:11:40,220 --> 00:11:41,120 What's an interval? 146 00:11:43,900 --> 00:11:46,660 It's a set of numbers between two numbers. 147 00:11:48,340 --> 00:11:50,580 We usually use interval notation. 148 00:11:51,660 --> 00:11:53,340 This case on an open interval. 149 00:11:54,440 --> 00:11:58,040 if it's continuous at every value on that interval. 150 00:12:01,360 --> 00:12:03,560 If it's continuous over all real numbers, 151 00:12:03,680 --> 00:12:05,020 we say it's continuous everywhere. 152 00:12:09,440 --> 00:12:12,720 So this function is continuous on this open interval 153 00:12:14,600 --> 00:12:16,640 and on this open interval. 154 00:12:20,240 --> 00:12:23,560 We say it's continuous on this set of numbers. 155 00:12:24,620 --> 00:12:29,265 It's continuous on every value inside this open interval, every value inside this 156 00:12:29,265 --> 00:12:30,015 open interval. 157 00:12:29,980 --> 00:12:35,800 What is the limit as x approaches 4 from the left? 158 00:12:36,020 --> 00:12:39,200 Let's call this 5. 159 00:12:39,200 --> 00:12:50,920 The limit as x approaches 4 from the left? 5. And what is f of 4? 5. 160 00:12:52,200 --> 00:12:59,300 Okay, if these are the same numbers, the function is definitely discontinuous at 4, 161 00:12:59,920 --> 00:13:05,940 however we say it's continuous from the left, because we went from the left. 162 00:13:08,720 --> 00:13:14,360 So it is continuous from the left because the function value equals that limit from 163 00:13:14,360 --> 00:13:15,260 the left. 164 00:13:15,540 --> 00:13:18,990 It is discontinuous from the right because the limit exists from the right but is 165 00:13:18,990 --> 00:13:19,740 not 166 00:13:19,220 --> 00:13:20,160 equal to the function value. 167 00:13:21,960 --> 00:13:32,220 So the definition being continuous from the left to the right is what I just said. 168 00:13:32,220 --> 00:13:34,040 No, that's not it. 169 00:13:35,420 --> 00:13:37,827 The function is continuous from the left at x equals a if the limit as x approaches 170 00:13:37,827 --> 00:13:38,960 a from the left equals f of a. 171 00:13:40,820 --> 00:14:01,000 f of x continuous on the left if the limit as x approaches at x equals a if f of x. 172 00:14:01,000 --> 00:14:03,420 equals f of a. 173 00:14:05,000 --> 00:14:07,200 f of a. 174 00:14:16,620 --> 00:14:19,680 The function is continuous from the right at x equals a if the limit as x approaches 175 00:14:19,680 --> 00:14:21,120 a from the right equals f of a. 176 00:14:22,700 --> 00:14:25,360 from the right 177 00:14:29,500 --> 00:14:38,820 at x equals a if the limit as x approaches a from the right 178 00:14:40,260 --> 00:14:42,400 equals F of. 179 00:14:42,400 --> 00:14:44,220 Let's call this fx. 180 00:15:04,520 --> 00:15:13,980 What is the limit as x approaches negative 2 of f of x? 181 00:15:19,320 --> 00:15:24,520 as x approaches 2, is it equal to 1? 182 00:15:25,780 --> 00:15:27,460 No. The answer is no. Why not? 183 00:15:29,660 --> 00:15:31,280 Because this limit 184 00:15:31,280 --> 00:15:34,560 as x approaches negative 2 185 00:15:34,560 --> 00:15:38,360 from the left does not exist. 186 00:15:39,460 --> 00:15:41,320 You can't approach it from the left. 187 00:15:42,540 --> 00:15:44,980 So this does not exist. 188 00:15:49,680 --> 00:15:53,720 The limit as x approaches negative 2 from the right, 189 00:15:54,620 --> 00:15:56,060 what does that equal? One. 190 00:15:57,940 --> 00:15:58,840 Okay. 191 00:16:03,560 --> 00:16:06,640 And what is the limit as x approaches 192 00:16:06,640 --> 00:16:10,180 4 from the left of f? 193 00:16:11,100 --> 00:16:20,260 5. The limit as x approaches 4 from the right does not exist. 194 00:16:20,260 --> 00:16:24,320 And the limit as x approaches 4. 195 00:16:25,340 --> 00:16:26,280 Same as x. 196 00:16:28,500 --> 00:16:30,580 Does not exist. 197 00:16:34,260 --> 00:16:35,160 OK. 198 00:16:42,840 --> 00:16:46,160 So is this function discontinuous at negative 2? 199 00:16:50,360 --> 00:16:51,760 It's a tricky question. 200 00:16:53,120 --> 00:16:54,020 How about a zero? 201 00:16:54,040 --> 00:16:54,940 Is it continuously zero? 202 00:16:56,140 --> 00:17:02,780 Is it continuous on every, or on this open interval? 203 00:17:04,400 --> 00:17:05,300 For sure. 204 00:17:06,740 --> 00:17:19,422 Ok, so, it's discontinuous at negative 2 because the limit, this limit does not 205 00:17:19,422 --> 00:17:22,140 equal f of 206 00:17:22,140 --> 00:17:23,080 negative 2. 207 00:17:26,020 --> 00:17:28,520 However, that's an end point, right? 208 00:17:28,860 --> 00:17:29,860 We have two end points. 209 00:17:31,020 --> 00:17:34,360 So, we define what it means to be continuous. 210 00:17:34,720 --> 00:17:42,056 We actually say this is continuous on the closed interval, even though it's not 211 00:17:42,056 --> 00:17:42,806 continuous 212 00:17:42,580 --> 00:17:43,480 at negative two. 213 00:17:44,140 --> 00:17:52,752 But we say it's continuous on the closed interval if the left endpoint is continuous 214 00:17:52,752 --> 00:17:53,900 from the 215 00:17:54,640 --> 00:17:55,740 Where do we have? 216 00:17:57,380 --> 00:18:03,586 If it's continuous from the right of the left endpoint, and if it's continuous from 217 00:18:03,586 --> 00:18:04,336 the 218 00:18:04,000 --> 00:18:09,860 left, the right endpoint, we say it's continuous on the closed interval. 219 00:18:13,200 --> 00:18:23,620 Okay, so this is the actual, so a function is continuous on the closed interval a to 220 00:18:23,620 --> 00:18:24,520 b. 221 00:18:25,400 --> 00:18:27,940 It has to be continuous everywhere inside the interval. 222 00:18:32,520 --> 00:18:38,848 And the limit as x approaches a, the left endpoint, from the right is equal to the 223 00:18:38,848 --> 00:18:39,598 function 224 00:18:39,220 --> 00:18:43,960 and B, the right endpoint, continuous from the left. 225 00:18:46,320 --> 00:18:50,360 Let me say it's continuous on that closed interval. 226 00:18:50,360 --> 00:18:52,260 Why do endpoint conditions matter? 227 00:18:57,380 --> 00:18:58,280 Well... 228 00:19:03,900 --> 00:19:05,620 Are we done, right? 229 00:19:22,220 --> 00:19:24,280 This is the square root of x, right? 230 00:19:28,820 --> 00:19:30,780 Is it continuous at x equals zero? 231 00:19:30,780 --> 00:19:38,680 No, because the limit as x approaches zero from the left doesn't exist. 232 00:19:39,780 --> 00:19:45,920 However, what interval is it continuous on? 233 00:19:51,780 --> 00:20:02,449 including zero because it's a left endpoint and as x approaches zero from the right 234 00:20:02,449 --> 00:20:03,199 does 235 00:20:03,160 --> 00:20:04,180 equal the function value. 236 00:20:04,940 --> 00:20:10,696 So it is continuous on this closed interval which is not talking about continuous at 237 00:20:10,696 --> 00:20:11,446 a 238 00:20:11,080 --> 00:20:11,980 point. 239 00:20:13,380 --> 00:20:17,898 It's as long as the left endpoint is continuous from the right, the right endpoint 240 00:20:17,898 --> 00:20:18,648 is continuous 241 00:20:18,500 --> 00:20:19,400 from the left. 242 00:20:20,020 --> 00:20:21,380 It's continuous on this interval. 243 00:20:21,380 --> 00:20:24,840 So what is the domain of this? 244 00:20:29,920 --> 00:20:31,180 Including zero, correct? 245 00:20:32,240 --> 00:20:39,380 Okay, so this is continuous on its domain. 246 00:20:42,760 --> 00:20:46,080 So in fact, every function we know 247 00:20:52,080 --> 00:20:55,120 is continuous on its domain. 248 00:20:56,100 --> 00:20:58,140 Every polynomial, every rational function, 249 00:20:58,260 --> 00:20:59,820 every radical function, every trigonometric, 250 00:20:59,820 --> 00:21:05,480 every exponential, every logarithmic function is continuous on its domain. 251 00:21:06,700 --> 00:21:08,760 If you know the domain, you know where it's continuous. 252 00:21:17,540 --> 00:21:22,761 You don't have to write, well, you can just say all functions we know are continuous 253 00:21:22,761 --> 00:21:23,740 on its domain. 254 00:21:23,740 --> 00:21:25,900 And then we've got a couple rules. 255 00:21:27,240 --> 00:21:29,300 Well, not rules, but properties. 256 00:21:32,780 --> 00:21:36,840 Any number times a continuous function is continuous. 257 00:21:39,540 --> 00:21:43,000 Seven times the square root of x is continuous on its domain. 258 00:21:44,220 --> 00:21:46,380 Any number times a continuous function is continuous. 259 00:21:46,380 --> 00:21:50,660 The sum or difference of continuous functions is continuous. 260 00:21:53,340 --> 00:22:01,640 Let's say g of x is sine of x plus x cubed or something. 261 00:22:02,300 --> 00:22:06,948 We know this is continuous because we know this is continuous and this is 262 00:22:06,948 --> 00:22:07,698 continuous. 263 00:22:07,880 --> 00:22:10,980 And the sum of continuous functions is continuous. 264 00:22:12,140 --> 00:22:14,440 The product of continuous functions is continuous. 265 00:22:18,280 --> 00:22:23,460 cosine of x times e to the x is continuous function. 266 00:22:24,140 --> 00:22:26,260 Trigonometric functions and exponential functions are continuous on their domains. 267 00:22:27,380 --> 00:22:29,220 All exponential functions are continuous. 268 00:22:29,580 --> 00:22:32,200 The product of continuous functions are continuous. 269 00:22:35,260 --> 00:22:44,780 The quotient is continuous is whenever g of any value is not equal to zero. 270 00:22:45,620 --> 00:22:46,820 And it won't be continuous there. 271 00:22:48,820 --> 00:22:53,952 But in scalar multiple any coefficient times continuous function, sum or difference, 272 00:22:53,952 --> 00:22:54,702 product 273 00:22:54,380 --> 00:22:57,980 of continuous functions and quotient are continuous. 274 00:22:57,980 --> 00:23:01,000 Where is this continuous? 275 00:23:07,540 --> 00:23:09,160 Is x to the fifth plus one continuous? 276 00:23:10,320 --> 00:23:11,220 No. 277 00:23:12,400 --> 00:23:15,300 It's a polynomial for sure continuous everywhere. 278 00:23:17,060 --> 00:23:18,260 Is e to the x continuous? 279 00:23:20,920 --> 00:23:22,900 Yeah, all exponential functions are continuous everywhere. 280 00:23:25,360 --> 00:23:27,160 Will e to the x ever be zero? 281 00:23:31,640 --> 00:23:34,680 All exponential functions always give you a positive value. 282 00:23:34,680 --> 00:23:37,000 since this will never be zero, 283 00:23:38,000 --> 00:23:39,980 this is definitely a continuous function 284 00:23:40,840 --> 00:23:42,240 over all real numbers. 285 00:23:43,720 --> 00:23:45,920 So we're looking for whether a function is continuous 286 00:23:45,920 --> 00:23:50,520 or not, especially with the quotients. 287 00:23:51,700 --> 00:23:53,480 You have to check, is this continuous? 288 00:23:53,660 --> 00:23:54,560 Where is this continuous? 289 00:23:54,640 --> 00:23:55,540 Where is this continuous? 290 00:23:55,980 --> 00:23:57,700 And then a special case to quotients, 291 00:23:58,340 --> 00:23:59,500 we gotta make sure it's not zero. 292 00:24:00,120 --> 00:24:02,600 Whenever the denominator is zero, it's not continuous. 293 00:24:02,600 --> 00:24:04,520 It's discontinuous. 294 00:24:07,220 --> 00:24:09,980 And then one more property. 295 00:24:12,100 --> 00:24:17,540 The composition of continuous functions are continuous. 296 00:24:22,300 --> 00:24:24,880 Just write that. 297 00:24:25,080 --> 00:24:27,580 The composition of continuous functions are continuous. 298 00:24:27,580 --> 00:24:31,680 F composed with G as long as F and G are continuous 299 00:24:32,520 --> 00:24:34,160 F of G is continuous 300 00:24:34,160 --> 00:24:35,060 Example... 301 00:24:34,600 --> 00:24:43,098 I have no idea what this function looks like, but do we know whether it is 302 00:24:43,098 --> 00:24:44,160 continuous or 303 00:24:44,160 --> 00:24:45,060 not? 304 00:24:47,560 --> 00:24:50,000 Is sine of x continuous over all real numbers? 305 00:24:51,340 --> 00:24:55,540 Yes, is 2 to the x continuous over all real numbers? Yes. 306 00:24:56,620 --> 00:24:58,380 So I plugged in sine of x 307 00:24:58,980 --> 00:25:00,400 into this function 308 00:25:01,300 --> 00:25:03,880 with guarantees continuous over all real numbers. 309 00:25:05,620 --> 00:25:06,520 Okay. 310 00:25:05,720 --> 00:25:08,140 Okay, let's do some examples. 311 00:25:08,140 --> 00:25:11,640 Let's see how far we go. 312 00:25:11,640 --> 00:25:13,820 I think we did 313 00:25:13,820 --> 00:25:15,280 a similar example. 314 00:25:15,440 --> 00:25:16,660 Let's do this. 315 00:25:25,280 --> 00:25:26,180 Okay. 316 00:25:27,780 --> 00:25:30,920 Actually, let me do this plus one. 317 00:25:33,520 --> 00:25:41,060 And then, e to the x, get the x to the first one. 318 00:25:41,060 --> 00:25:45,600 So where, or on what, where is this continuous? 319 00:25:56,100 --> 00:25:57,380 So how are we going to do this? 320 00:26:03,660 --> 00:26:07,860 So, we have to check, so any piecewise functions, we have to check each piece 321 00:26:07,860 --> 00:26:08,610 separately. 322 00:26:10,280 --> 00:26:11,180 Okay. 323 00:26:13,840 --> 00:26:26,609 So if x is greater than or equal, let's go with greater than zero, greater than pi, 324 00:26:26,609 --> 00:26:27,360 then 325 00:26:27,360 --> 00:26:31,480 f of x equals sine of x, right? 326 00:26:36,440 --> 00:26:37,520 Is sine of x continuous? 327 00:26:39,620 --> 00:26:40,520 Definitely. 328 00:26:41,540 --> 00:26:44,120 OK, so we definitely know it's continuous from pi to infinity. 329 00:26:48,340 --> 00:27:12,283 For zero less than or equal to x and x less than pi, f of x equals x squared plus 330 00:27:12,283 --> 00:27:13,480 one. 331 00:27:18,240 --> 00:27:31,407 Then if x is less than zero, f of x equals e to the x, which is an exponential 332 00:27:31,407 --> 00:27:32,157 function, 333 00:27:32,840 --> 00:27:33,740 continuous everywhere. 334 00:27:35,680 --> 00:27:38,760 We know it's continuous here. 335 00:27:39,300 --> 00:27:45,455 So we check each piece. As long as they are continuous, it's continuous on the open 336 00:27:45,455 --> 00:27:46,205 interval. 337 00:27:47,580 --> 00:27:53,460 So really all we have to do is check at pi and at zero. 338 00:28:03,520 --> 00:28:05,580 at x equals zero 339 00:28:09,140 --> 00:28:11,200 okay this is where it changes 340 00:28:12,440 --> 00:28:16,560 from this function to this function 341 00:28:18,520 --> 00:28:21,700 we have to do the limit from the left and the limit from the right separately 342 00:28:21,700 --> 00:28:28,880 So what is the limit as x approaches... 343 00:28:28,880 --> 00:28:31,500 uh... sorry, let's go to pi first, my bad. 344 00:28:33,020 --> 00:28:34,680 pi from the left. 345 00:28:43,540 --> 00:28:45,160 So from the left, 346 00:28:45,740 --> 00:28:47,240 f of x is equal to this. 347 00:28:48,300 --> 00:28:49,540 So this is the limit 348 00:28:49,540 --> 00:28:54,820 as x approaches pi from the left of x squared plus one. 349 00:28:56,700 --> 00:29:01,660 And what does that equal? Direct substitution of pi squared plus one. 350 00:29:02,880 --> 00:29:03,780 Sum number. 351 00:29:07,760 --> 00:29:13,360 The limit as x approaches pi from the right. 352 00:29:22,160 --> 00:29:28,320 limit as x approaches pi from the right. 353 00:29:28,900 --> 00:29:32,400 So f of x is equal to sine of x. 354 00:29:32,560 --> 00:29:40,740 which equals sine of pi which equals 355 00:29:44,640 --> 00:29:47,460 zero thank you okay 356 00:29:50,160 --> 00:29:52,960 so what is the limit as x approaches pi 357 00:29:57,610 --> 00:30:01,130 It does not exist. 358 00:30:08,930 --> 00:30:13,290 Is this function continuous from the left or the right or neither at pi? 359 00:30:19,150 --> 00:30:20,790 What is f of pi? 360 00:30:25,670 --> 00:30:30,050 Pi is here, so it's sine of pi, which is zero. 361 00:30:33,710 --> 00:30:35,990 So is it continuous from the left or the right? 362 00:30:38,230 --> 00:30:39,130 At pi. 363 00:30:40,870 --> 00:30:44,010 From the right. So, final with this. 364 00:30:50,610 --> 00:30:52,250 Our function looks like this. 365 00:30:53,930 --> 00:30:55,650 So it is continuous on the right. 366 00:30:59,590 --> 00:31:00,490 What is it? 367 00:31:00,270 --> 00:31:03,010 x squared plus one. Looks like that. 368 00:31:03,890 --> 00:31:09,152 sorry, open circle here, is discontinuous on the left, and it's definitely 369 00:31:09,152 --> 00:31:09,902 discontinuous 370 00:31:09,590 --> 00:31:10,490 at pi. 371 00:31:12,570 --> 00:31:14,210 Now let's check at zero. 372 00:31:15,490 --> 00:31:25,748 The limit as x approaches zero from the left, the limit as x approaches zero from 373 00:31:25,748 --> 00:31:27,030 the left, 374 00:31:27,030 --> 00:31:33,010 From the left, f of x is right here. 375 00:31:33,990 --> 00:31:35,890 x squared plus 1. 376 00:31:41,150 --> 00:31:45,290 Direct substitution plus 1. 377 00:31:49,950 --> 00:31:50,850 Question? 378 00:31:54,110 --> 00:31:55,750 Oh, I'm sorry. 379 00:31:55,750 --> 00:32:03,979 I was looking at the wrong thing. This is 0 from the left. This is e to the x. My 380 00:32:03,979 --> 00:32:04,729 bad. 381 00:32:11,410 --> 00:32:22,750 Sorry, this is e to the x, which is e to the 0, which is 1. So the limit as x 382 00:32:22,750 --> 00:32:23,500 approaches 383 00:32:23,290 --> 00:32:33,010 0 from the right, the limit as x approaches 0 from the right of, this was the x 384 00:32:33,010 --> 00:32:33,760 squared 385 00:32:33,550 --> 00:32:47,779 plus 1, which is 0 squared plus 1, which is 1. So, the limit as x approaches 0 is 1. 386 00:32:47,779 --> 00:32:48,529 And 387 00:32:48,490 --> 00:32:59,970 f of zero is this guy, zero squared plus one. 388 00:33:04,490 --> 00:33:13,550 So it's definitely continuous at zero. 389 00:33:13,550 --> 00:33:18,890 Ok, any questions on this? 390 00:33:19,590 --> 00:33:22,970 So for piecewise we just gotta check, the main thing we gotta do is check where the 391 00:33:22,970 --> 00:33:24,650 two functions meet. 392 00:33:25,210 --> 00:33:27,590 And then of course inside you have to check where is the continuous. 393 00:33:27,590 --> 00:33:29,570 I chose three continuous functions. 394 00:33:29,570 --> 00:33:33,830 OK, the intermediate value theorem. 395 00:33:35,730 --> 00:33:37,570 Last quick, this is a quick topic. 396 00:33:38,430 --> 00:33:39,730 Let me find it. 397 00:33:39,730 --> 00:33:40,670 Shh. 398 00:33:40,670 --> 00:33:43,110 Okay, write this down and then we'll talk later. 399 00:33:43,110 --> 00:33:44,510 Okay. 400 00:33:44,510 --> 00:33:47,430 Okay, so this is called an existence theorem. 401 00:33:48,690 --> 00:33:55,470 We're gonna have three, sort of four different existence theorems in calculus. 402 00:33:56,930 --> 00:34:00,090 All it does is tell you a value exists. 403 00:34:01,050 --> 00:34:03,090 It doesn't help us find the value at all. 404 00:34:03,570 --> 00:34:05,990 It just guarantees for it to exist. 405 00:34:06,630 --> 00:34:10,142 This is an existence theorem: it guarantees a value exists without telling us how to 406 00:34:10,142 --> 00:34:10,892 find it. 407 00:34:10,930 --> 00:34:12,030 Anybody remember this? 408 00:34:13,470 --> 00:34:18,290 But technically we don't really have it till this class. 409 00:34:18,370 --> 00:34:20,010 Why is that? 410 00:34:26,930 --> 00:34:28,770 Because we don't need it, somebody said. 411 00:34:30,990 --> 00:34:31,890 This is not true. 412 00:34:31,850 --> 00:34:33,330 We definitely need it. 413 00:34:48,150 --> 00:34:53,610 Here we go. 414 00:34:55,410 --> 00:34:57,510 What about this has to do with calculus? 415 00:35:00,750 --> 00:35:03,490 I don't see anything about limits in here. 416 00:35:05,170 --> 00:35:06,110 Those are good. 417 00:35:06,350 --> 00:35:07,690 Well, their sort is. 418 00:35:10,490 --> 00:35:11,390 OK. 419 00:35:12,470 --> 00:35:22,974 The only hypothesis, hypothesis C, is f has to be a continuous function on a closed 420 00:35:22,974 --> 00:35:23,724 interval. 421 00:35:27,330 --> 00:35:31,650 And truly we didn't define what it means to be continuous till just today. 422 00:35:32,750 --> 00:35:36,781 Before your teacher said, well you can draw it without lifting up your pencil, it's 423 00:35:36,781 --> 00:35:37,531 continuous. 424 00:35:38,150 --> 00:35:45,486 But truly right, it's continuous on a closed interval, means the limit as x 425 00:35:45,486 --> 00:35:46,236 approaches 426 00:35:47,590 --> 00:35:50,370 some other value, let me call it C, 427 00:35:51,350 --> 00:35:56,468 Continuity on the closed interval includes continuity from the right at a and from 428 00:35:56,468 --> 00:35:57,930 the left at b. 429 00:35:58,590 --> 00:36:00,430 for every value on the open interval. 430 00:36:01,430 --> 00:36:05,530 And it's continuous from the left at b, 431 00:36:05,790 --> 00:36:06,970 and it's continuous from the right at a. 432 00:36:07,890 --> 00:36:10,370 Okay, we now know the actual definition. 433 00:36:10,950 --> 00:36:14,670 So that's why it's part of this class, okay? 434 00:36:14,670 --> 00:36:28,062 But all this thing says is, as long as you have a continuous function, let's put in 435 00:36:28,062 --> 00:36:28,850 some 436 00:36:28,850 --> 00:36:40,210 numbers for class one and eight or something, two and ten. 437 00:36:40,210 --> 00:36:48,912 as long as we have a continuous function, let n be any number between f of a and f 438 00:36:48,912 --> 00:36:49,662 of 439 00:36:49,370 --> 00:36:59,030 b. What is f of b? 10. f of a is 2. 440 00:37:01,650 --> 00:37:06,730 n can be any number between 10 and 2. Any number here. 441 00:37:16,430 --> 00:37:23,707 And then we are guaranteed, actually as long as f of a is not equal f of b, that's 442 00:37:23,707 --> 00:37:24,457 the 443 00:37:24,090 --> 00:37:25,790 one thing, it's definitely not equal. 444 00:37:27,210 --> 00:37:35,990 There must exist a value c, there must be a c for any number, let's call this number 445 00:37:35,990 --> 00:37:51,590 n there must be a C in the open interval such that F of C equals n. So for example, 446 00:37:58,010 --> 00:38:04,490 5, are we guaranteed to have a value between 1 and 8? 447 00:38:05,110 --> 00:38:06,450 Yes, we are guaranteed. 448 00:38:06,730 --> 00:38:15,559 Because 5 is between 10 and 2, we are guaranteed a value in 1 to 8, the open 449 00:38:15,559 --> 00:38:16,309 interval. 450 00:38:18,010 --> 00:38:20,030 such an F of C equals five. 451 00:38:21,510 --> 00:38:24,030 Our function takes on every single value 452 00:38:24,030 --> 00:38:26,470 and sometimes it can take on it twice, right? 453 00:38:26,490 --> 00:38:30,210 So in fact, there are three different C values, 454 00:38:31,390 --> 00:38:32,930 C one, C two, C three, 455 00:38:34,070 --> 00:38:36,390 but this theorem intermediate value theorem 456 00:38:36,390 --> 00:38:38,670 just guarantees there's at least one. 457 00:38:40,530 --> 00:38:42,530 When is this used? 458 00:38:42,530 --> 00:38:50,530 And typically something like this. 459 00:38:54,720 --> 00:39:00,260 Show where is a 460 00:39:04,480 --> 00:39:12,660 Show that f has a zero between zero and two. 461 00:39:28,900 --> 00:39:30,980 Now what's a zero of a function? 462 00:39:34,860 --> 00:39:37,340 A zero of a function is an x value. 463 00:39:38,460 --> 00:39:53,160 Okay, a zero is an x value that makes f of x equal to zero. 464 00:39:57,080 --> 00:40:01,240 So to show that, we're going to use the intermediate value term. 465 00:40:01,920 --> 00:40:12,080 Since f is a polynomial, it is continuous on the closed interval from zero to two. 466 00:40:12,300 --> 00:40:13,460 And why is that? 467 00:40:20,980 --> 00:40:23,520 How do we know this is continuous on the closed interval? 468 00:40:25,620 --> 00:40:35,220 because it's a polynomial, are continuous everywhere. 469 00:40:41,720 --> 00:40:45,760 And what is f of zero equal to? 470 00:40:47,380 --> 00:40:50,140 Zero cubed minus four, negative four. 471 00:40:50,140 --> 00:40:58,560 F of 2 is 2 cubed minus 4 is 4. 472 00:41:01,900 --> 00:41:23,412 By the Intermediate Value Theorem, there exists a c in the open interval from zero 473 00:41:23,412 --> 00:41:26,280 to two. 474 00:41:26,280 --> 00:41:31,520 the open interval such that 475 00:41:34,280 --> 00:41:38,080 f of c equals zero because zero 476 00:41:38,080 --> 00:41:43,140 is between negative four 477 00:41:43,140 --> 00:41:44,040 and four 478 00:41:53,660 --> 00:41:59,280 This function looks something like this, right? 479 00:42:01,360 --> 00:42:06,200 This was negative four. 480 00:42:09,240 --> 00:42:10,700 This was positive four. 481 00:42:12,600 --> 00:42:16,480 Our function since it's continuous it takes on every single value here. 482 00:42:19,220 --> 00:42:21,860 And this would be our value of C right there. 483 00:42:22,320 --> 00:42:23,580 Such a F of C is zero. 484 00:42:24,660 --> 00:42:25,560 Okay. 485 00:42:25,680 --> 00:42:28,910 We're only going to use it a few times but sometimes we're going to have to mention 486 00:42:28,910 --> 00:42:29,660 the 487 00:42:29,100 --> 00:42:30,080 intermediate value theorem. 488 00:42:31,480 --> 00:42:32,380 And so on. 489 00:42:32,180 --> 00:42:33,180 Okay we're going to stop here.