
Title: Some Continuous Functions

Series: Real Analysis

YouTubeTitle: Real Analysis 33  Some Continuous Functions

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Subtitle in English
1 00:00:00,430 –> 00:00:02,329 Hello and welcome back to
2 00:00:02,339 –> 00:00:03,880 real analysis.
3 00:00:04,570 –> 00:00:05,969 And first many, many thanks
4 00:00:05,980 –> 00:00:07,179 to all the nice supporters
5 00:00:07,190 –> 00:00:08,670 on steady or paypal.
6 00:00:09,250 –> 00:00:10,630 Now, in this video today,
7 00:00:10,640 –> 00:00:11,840 you will get to know some
8 00:00:11,850 –> 00:00:13,180 important continuous
9 00:00:13,189 –> 00:00:13,869 functions.
10 00:00:14,630 –> 00:00:16,180 Some of them are so important
11 00:00:16,190 –> 00:00:17,319 that they are often called
12 00:00:17,329 –> 00:00:18,780 elementary functions.
13 00:00:19,489 –> 00:00:21,159 Indeed, all of them we will
14 00:00:21,170 –> 00:00:22,620 consider again later when
15 00:00:22,629 –> 00:00:24,040 we differentiate and
16 00:00:24,049 –> 00:00:24,899 integrate.
17 00:00:25,440 –> 00:00:26,579 Hence, let’s immediately
18 00:00:26,590 –> 00:00:28,000 start with the most important
19 00:00:28,010 –> 00:00:28,420 one.
20 00:00:28,430 –> 00:00:29,879 The exponential function
21 00:00:30,540 –> 00:00:31,709 we shorten the name with
22 00:00:31,719 –> 00:00:33,669 X and the domain is the whole
23 00:00:33,680 –> 00:00:34,700 real number line.
24 00:00:35,119 –> 00:00:36,369 And we have already learned
25 00:00:36,380 –> 00:00:37,759 in the former video that
26 00:00:37,770 –> 00:00:39,619 it is defined by a series,
27 00:00:40,400 –> 00:00:42,029 namely we have X to the power
28 00:00:42,040 –> 00:00:43,540 K divided by K
29 00:00:43,549 –> 00:00:44,580 factorial.
30 00:00:45,290 –> 00:00:46,669 And now you might already
31 00:00:46,680 –> 00:00:48,509 know at the point X is equal
32 00:00:48,520 –> 00:00:49,810 to one, we find the
33 00:00:49,819 –> 00:00:51,540 socalled Euler number.
34 00:00:52,330 –> 00:00:53,830 Also it’s possible to show
35 00:00:53,840 –> 00:00:55,389 that this number can be defined
36 00:00:55,400 –> 00:00:56,630 by another sequence
37 00:00:57,380 –> 00:00:59,000 namely by the limit one
38 00:00:59,009 –> 00:01:00,599 plus one over N to the
39 00:01:00,610 –> 00:01:02,430 power N which is
40 00:01:02,439 –> 00:01:02,939 roughly
41 00:01:02,950 –> 00:01:04,910 2.718.
42 00:01:05,559 –> 00:01:07,040 Now this exponential function
43 00:01:07,050 –> 00:01:08,779 shouldn’t be new to you because
44 00:01:08,790 –> 00:01:10,099 we have already discussed
45 00:01:10,110 –> 00:01:11,500 it in part 22
46 00:01:12,099 –> 00:01:13,260 there, we have shown the
47 00:01:13,269 –> 00:01:14,860 fundamental multiplicative
48 00:01:14,870 –> 00:01:16,580 identity for the exponential
49 00:01:16,589 –> 00:01:18,000 function which
50 00:01:18,010 –> 00:01:19,339 translates the addition
51 00:01:19,349 –> 00:01:20,489 inside into a
52 00:01:20,500 –> 00:01:22,120 multiplication outside.
53 00:01:22,580 –> 00:01:23,879 And because we have this
54 00:01:23,889 –> 00:01:25,559 property, we can immediately
55 00:01:25,569 –> 00:01:26,680 explain the name
56 00:01:26,690 –> 00:01:28,559 exponential for this function.
57 00:01:29,160 –> 00:01:30,989 For example, if we have X
58 00:01:31,000 –> 00:01:32,870 of two, we can
59 00:01:32,879 –> 00:01:34,669 write this as X of one
60 00:01:34,680 –> 00:01:35,650 plus one
61 00:01:36,430 –> 00:01:37,870 which is by our fundamental
62 00:01:37,879 –> 00:01:39,489 multiplicative identity here
63 00:01:39,500 –> 00:01:41,300 exactly X of one
64 00:01:41,309 –> 00:01:42,580 times X of one.
65 00:01:43,110 –> 00:01:44,419 So it’s Euler number
66 00:01:44,430 –> 00:01:45,209 squared.
67 00:01:45,849 –> 00:01:47,790 So you see the left here
68 00:01:47,800 –> 00:01:49,400 and the right hand side are
69 00:01:49,410 –> 00:01:49,989 the same.
70 00:01:50,769 –> 00:01:52,209 And indeed, we can show that
71 00:01:52,220 –> 00:01:53,699 in general, the exponential
72 00:01:53,709 –> 00:01:55,430 function X of X is
73 00:01:55,440 –> 00:01:57,239 exactly Euler number to the
74 00:01:57,250 –> 00:01:57,989 power X.
75 00:01:58,739 –> 00:01:59,959 Therefore, the name makes
76 00:01:59,970 –> 00:02:01,529 sense because we can rewrite
77 00:02:01,540 –> 00:02:03,019 a function such that we have
78 00:02:03,029 –> 00:02:04,139 the exponent here.
79 00:02:04,849 –> 00:02:05,980 And of course, we can show
80 00:02:05,989 –> 00:02:07,750 even more properties of this
81 00:02:07,760 –> 00:02:09,029 important function here.
82 00:02:09,839 –> 00:02:10,860 The first part might not
83 00:02:10,869 –> 00:02:11,779 surprise you.
84 00:02:11,789 –> 00:02:13,029 The exponential function
85 00:02:13,039 –> 00:02:14,619 is a continuous function.
86 00:02:15,119 –> 00:02:16,820 So it’s continuous at all
87 00:02:16,830 –> 00:02:17,960 points in R.
88 00:02:18,710 –> 00:02:20,149 Then the next property is
89 00:02:20,160 –> 00:02:21,580 that this function is at
90 00:02:21,589 –> 00:02:23,289 all points increasing
91 00:02:23,570 –> 00:02:24,750 more concretely, we would
92 00:02:24,759 –> 00:02:26,300 say the exponential function
93 00:02:26,309 –> 00:02:27,960 is strictly monotonically
94 00:02:27,970 –> 00:02:29,929 increasing, which simply
95 00:02:29,940 –> 00:02:31,699 means if we look at two points
96 00:02:31,710 –> 00:02:33,500 X and Y where Y is
97 00:02:33,509 –> 00:02:35,369 greater than X, then also
98 00:02:35,380 –> 00:02:36,990 the values fulfill the same
99 00:02:37,000 –> 00:02:37,889 inequality.
100 00:02:38,139 –> 00:02:40,029 So X of Y is greater
101 00:02:40,039 –> 00:02:41,229 than X of X.
102 00:02:41,800 –> 00:02:43,070 In addition, we can also
103 00:02:43,080 –> 00:02:44,610 say what happens in the limits
104 00:02:44,619 –> 00:02:46,559 X to infinity and X
105 00:02:46,570 –> 00:02:47,929 to minus infinity
106 00:02:48,100 –> 00:02:49,889 for plus infinity, we
107 00:02:49,899 –> 00:02:51,369 exceed all bounds.
108 00:02:51,380 –> 00:02:52,960 Therefore, we go to plus
109 00:02:52,970 –> 00:02:53,649 infinity.
110 00:02:54,410 –> 00:02:56,119 However, for minus infinity,
111 00:02:56,130 –> 00:02:57,360 it’s different because we
112 00:02:57,369 –> 00:02:59,160 stay positive the whole time
113 00:02:59,169 –> 00:03:00,960 we reach zero in the limit.
114 00:03:01,429 –> 00:03:01,800 OK.
115 00:03:01,809 –> 00:03:03,110 So these are the important
116 00:03:03,119 –> 00:03:04,399 properties you really should
117 00:03:04,410 –> 00:03:06,270 remember for the exponential
118 00:03:06,279 –> 00:03:06,800 function.
119 00:03:07,369 –> 00:03:08,919 Now, one important conclusion
120 00:03:08,929 –> 00:03:10,320 here is what we will need
121 00:03:10,330 –> 00:03:11,669 for the next example.
122 00:03:12,250 –> 00:03:13,720 Namely now you are able to
123 00:03:13,729 –> 00:03:15,380 show if we change the codomain
124 00:03:15,389 –> 00:03:16,619 to zero
125 00:03:16,630 –> 00:03:17,509 infinity.
126 00:03:17,929 –> 00:03:19,830 So the interval of the positive
127 00:03:19,839 –> 00:03:21,679 numbers, then we
128 00:03:21,690 –> 00:03:23,119 get a bijective map.
129 00:03:23,740 –> 00:03:25,110 Indeed, that’s not hard to
130 00:03:25,119 –> 00:03:26,630 show when you know all the
131 00:03:26,639 –> 00:03:27,410 other things here.
132 00:03:28,089 –> 00:03:29,509 And of course, now you already
133 00:03:29,520 –> 00:03:31,490 know a bijective map has
134 00:03:31,500 –> 00:03:32,589 an inverse map.
135 00:03:33,050 –> 00:03:34,339 And this inverse is what
136 00:03:34,350 –> 00:03:36,149 we usually call the logarithm
137 00:03:36,160 –> 00:03:36,830 function.
138 00:03:37,240 –> 00:03:38,770 So you see this is a simple
139 00:03:38,779 –> 00:03:40,259 definition and then we can
140 00:03:40,270 –> 00:03:41,600 ask the question, what are
141 00:03:41,610 –> 00:03:43,190 the properties of the logarithm
142 00:03:43,199 –> 00:03:43,669 function?
143 00:03:44,300 –> 00:03:46,050 And of course, not so surprising,
144 00:03:46,059 –> 00:03:47,589 we can deduce a lot of
145 00:03:47,600 –> 00:03:49,360 properties from the exponential
146 00:03:49,369 –> 00:03:49,880 function.
147 00:03:50,179 –> 00:03:51,309 So maybe let’s look at the
148 00:03:51,320 –> 00:03:52,960 graphs to see what we already
149 00:03:52,970 –> 00:03:53,470 have.
150 00:03:54,350 –> 00:03:55,690 So in blue, you see the graph
151 00:03:55,699 –> 00:03:56,990 of the exponential function
152 00:03:57,050 –> 00:03:58,809 and then we reflect it where
153 00:03:58,820 –> 00:04:00,169 the mirror line is this
154 00:04:00,179 –> 00:04:01,690 diagonal here.
155 00:04:01,699 –> 00:04:03,449 Please recall this reflecting
156 00:04:03,460 –> 00:04:04,500 follows immediately by the
157 00:04:04,509 –> 00:04:06,089 definition of the graph and
158 00:04:06,100 –> 00:04:07,089 the inverse function
159 00:04:07,910 –> 00:04:08,250 HC.
160 00:04:08,259 –> 00:04:09,580 And we, you find the graph
161 00:04:09,589 –> 00:04:11,130 of the logarithm function.
162 00:04:11,550 –> 00:04:12,960 Now what we see is that the
163 00:04:12,970 –> 00:04:14,589 logarithm function is also
164 00:04:14,600 –> 00:04:15,990 strictly monotonically
165 00:04:16,000 –> 00:04:17,450 increasing and it should
166 00:04:17,459 –> 00:04:19,029 still be continuous.
167 00:04:19,559 –> 00:04:20,890 Indeed showing these two
168 00:04:20,899 –> 00:04:22,890 things is a very good exercise.
169 00:04:23,059 –> 00:04:24,700 First, you show the increasing
170 00:04:24,709 –> 00:04:25,910 part and then you use the
171 00:04:25,920 –> 00:04:27,519 epsilon delta criterion to
172 00:04:27,529 –> 00:04:28,799 show the continuity.
173 00:04:29,359 –> 00:04:29,859 OK.
174 00:04:29,869 –> 00:04:31,239 However, the most important
175 00:04:31,250 –> 00:04:32,459 property for the logarithm
176 00:04:32,470 –> 00:04:34,049 function follows immediately
177 00:04:34,059 –> 00:04:35,899 from the fundamental multiplicative
178 00:04:35,910 –> 00:04:37,690 identity for the exponential
179 00:04:37,700 –> 00:04:38,200 function.
180 00:04:38,720 –> 00:04:39,000 OK.
181 00:04:39,010 –> 00:04:40,320 Now, because the logarithm
182 00:04:40,329 –> 00:04:42,160 function is the inverse function,
183 00:04:42,170 –> 00:04:43,839 this identity goes the other
184 00:04:43,850 –> 00:04:44,720 way around.
185 00:04:45,309 –> 00:04:47,179 So the product inside gets
186 00:04:47,190 –> 00:04:48,739 translated to the sum
187 00:04:48,750 –> 00:04:49,500 outside.
188 00:04:50,160 –> 00:04:50,519 OK?
189 00:04:50,529 –> 00:04:52,200 I can tell you this identity
190 00:04:52,209 –> 00:04:53,380 is very important.
191 00:04:53,390 –> 00:04:54,760 So please remember it.
192 00:04:55,309 –> 00:04:56,690 Now, with this, we go to
193 00:04:56,700 –> 00:04:58,589 our next examples which are
194 00:04:58,600 –> 00:04:59,720 polynomials
195 00:05:00,350 –> 00:05:00,839 for them.
196 00:05:00,850 –> 00:05:01,829 I don’t have to tell you
197 00:05:01,839 –> 00:05:03,459 a lot because you already
198 00:05:03,470 –> 00:05:04,079 know them.
199 00:05:04,760 –> 00:05:05,959 The important part here is
200 00:05:05,970 –> 00:05:07,579 the domain of a polynomial
201 00:05:07,589 –> 00:05:08,970 is the whole real number
202 00:05:08,980 –> 00:05:10,910 line and defined as it by
203 00:05:10,920 –> 00:05:12,910 giving finitely many coefficients
204 00:05:12,920 –> 00:05:13,279 A.
205 00:05:13,950 –> 00:05:15,049 And in the case that the
206 00:05:15,059 –> 00:05:16,970 last one am here is
207 00:05:16,980 –> 00:05:17,950 nonzero.
208 00:05:17,959 –> 00:05:19,570 We say that the polynomial
209 00:05:19,579 –> 00:05:21,519 has to create M as a
210 00:05:21,529 –> 00:05:22,839 reminder, we have already
211 00:05:22,850 –> 00:05:24,239 showed that the polynomials
212 00:05:24,250 –> 00:05:25,970 are continuous functions.
213 00:05:26,549 –> 00:05:27,929 So nothing new here.
214 00:05:27,940 –> 00:05:29,890 But we need to know the polynomials
215 00:05:29,899 –> 00:05:31,450 to define the next class
216 00:05:31,459 –> 00:05:32,529 of examples
217 00:05:33,350 –> 00:05:33,660 here.
218 00:05:33,670 –> 00:05:35,029 Please recall for a
219 00:05:35,040 –> 00:05:36,600 polynomial and important
220 00:05:36,609 –> 00:05:38,309 ingredient was that we have
221 00:05:38,320 –> 00:05:39,809 a highest power M here.
222 00:05:40,279 –> 00:05:41,600 If we don’t want to have
223 00:05:41,609 –> 00:05:42,910 that, we get something we
224 00:05:42,920 –> 00:05:44,549 call a power series.
225 00:05:44,970 –> 00:05:46,660 In fact, this is a very general
226 00:05:46,670 –> 00:05:48,209 term where we already know
227 00:05:48,220 –> 00:05:49,220 one example,
228 00:05:49,790 –> 00:05:51,630 namely the exponential function
229 00:05:51,640 –> 00:05:53,540 from above was defined using
230 00:05:53,549 –> 00:05:54,640 a power series.
231 00:05:55,149 –> 00:05:56,880 Now, in general a power series
232 00:05:56,890 –> 00:05:58,540 does not need to be defined
233 00:05:58,549 –> 00:05:59,730 on the whole real number
234 00:05:59,739 –> 00:06:00,179 line.
235 00:06:00,589 –> 00:06:02,149 So we could have a smaller
236 00:06:02,160 –> 00:06:03,109 domain D.
237 00:06:03,850 –> 00:06:05,320 However, otherwise it looks
238 00:06:05,329 –> 00:06:06,750 like a polynomial where we
239 00:06:06,760 –> 00:06:08,579 have coefficients A K here.
240 00:06:08,609 –> 00:06:10,510 But now we could have infinitely
241 00:06:10,519 –> 00:06:11,040 many.
242 00:06:11,559 –> 00:06:12,880 In other words, for each
243 00:06:12,890 –> 00:06:14,799 sequence A K, we get out
244 00:06:14,809 –> 00:06:16,239 a function we call a power
245 00:06:16,250 –> 00:06:16,880 series.
246 00:06:17,570 –> 00:06:18,829 In this case, the domain
247 00:06:18,839 –> 00:06:20,399 of definition is given by
248 00:06:20,410 –> 00:06:21,940 all the points X in
249 00:06:21,950 –> 00:06:23,899 R that satisfy
250 00:06:23,910 –> 00:06:25,589 that this series is indeed
251 00:06:25,600 –> 00:06:26,809 a convergent one.
252 00:06:27,649 –> 00:06:28,790 And of course, you already
253 00:06:28,799 –> 00:06:30,269 know a lot of criteria you
254 00:06:30,279 –> 00:06:31,709 can use here to check for
255 00:06:31,720 –> 00:06:32,510 convergence.
256 00:06:33,040 –> 00:06:33,279 OK.
257 00:06:33,290 –> 00:06:34,489 Let’s immediately discuss
258 00:06:34,500 –> 00:06:36,329 a very famous example here
259 00:06:36,339 –> 00:06:37,899 which means we fix a special
260 00:06:37,910 –> 00:06:38,420 sequence.
261 00:06:38,429 –> 00:06:40,260 A K let’s start a
262 00:06:40,269 –> 00:06:41,549 sequence with the zeroth
263 00:06:41,559 –> 00:06:42,839 member which should be
264 00:06:42,850 –> 00:06:43,570 zero.
265 00:06:44,109 –> 00:06:45,679 And then I want one, but
266 00:06:45,690 –> 00:06:47,670 let’s write it as one divided
267 00:06:47,679 –> 00:06:49,010 by one factorial,
268 00:06:49,700 –> 00:06:51,339 then afterwards comes zero
269 00:06:51,350 –> 00:06:52,910 again and then
270 00:06:52,920 –> 00:06:54,720 minus one divided by
271 00:06:54,730 –> 00:06:56,230 three factorial.
272 00:06:56,809 –> 00:06:58,109 Then not surprising we have
273 00:06:58,119 –> 00:06:59,820 zero again and then one
274 00:06:59,829 –> 00:07:01,820 divided by five factorial,
275 00:07:02,230 –> 00:07:03,809 then zero and then
276 00:07:03,820 –> 00:07:05,570 minus one divided by
277 00:07:05,579 –> 00:07:06,820 seven factorial.
278 00:07:07,540 –> 00:07:07,829 OK.
279 00:07:07,839 –> 00:07:09,109 With this, I think you see
280 00:07:09,119 –> 00:07:10,440 what the whole wall is and
281 00:07:10,450 –> 00:07:11,829 that you can continue it.
282 00:07:12,679 –> 00:07:14,279 And now this sequence gives
283 00:07:14,290 –> 00:07:16,070 us a power series we usually
284 00:07:16,079 –> 00:07:17,899 call sine of X.
285 00:07:18,299 –> 00:07:19,779 Also, it’s not hard to check
286 00:07:19,790 –> 00:07:21,019 that the whole real number
287 00:07:21,029 –> 00:07:22,600 line is the domain D
288 00:07:23,019 –> 00:07:24,899 indeed, a lot of common functions
289 00:07:24,910 –> 00:07:26,339 can be defined with such
290 00:07:26,350 –> 00:07:27,420 a power series.
291 00:07:27,940 –> 00:07:29,299 And for this reason, I give
292 00:07:29,309 –> 00:07:31,119 you here a general result.
293 00:07:31,440 –> 00:07:33,160 Now, for any power series,
294 00:07:33,170 –> 00:07:34,529 given the coefficients A
295 00:07:34,540 –> 00:07:36,079 K, you find the
296 00:07:36,089 –> 00:07:38,000 maximum radius R
297 00:07:38,660 –> 00:07:40,500 where also infinity as a
298 00:07:40,510 –> 00:07:41,959 symbol is possible,
299 00:07:42,609 –> 00:07:43,720 which indeed would be the
300 00:07:43,730 –> 00:07:45,170 best case scenario.
301 00:07:45,700 –> 00:07:47,019 Now, this maximum radius
302 00:07:47,029 –> 00:07:48,910 fulfills that the open interval
303 00:07:48,920 –> 00:07:50,890 from minus R to R lies
304 00:07:50,899 –> 00:07:52,350 completely in the domain
305 00:07:52,359 –> 00:07:52,760 D.
306 00:07:53,329 –> 00:07:54,829 Now I can also tell you we
307 00:07:54,839 –> 00:07:55,989 immediately get that the
308 00:07:56,000 –> 00:07:57,700 power series is a continuous
309 00:07:57,709 –> 00:07:59,510 function on this interval.
310 00:07:59,670 –> 00:08:01,100 So in the best case scenario
311 00:08:01,109 –> 00:08:02,640 like for the sine function,
312 00:08:02,649 –> 00:08:04,239 we have a continuous function
313 00:08:04,250 –> 00:08:05,549 defined on the whole real
314 00:08:05,559 –> 00:08:06,320 number line.
315 00:08:06,910 –> 00:08:08,649 Now even better for the theorem,
316 00:08:08,660 –> 00:08:10,570 we have a formula to calculate
317 00:08:10,579 –> 00:08:11,690 this number R.
318 00:08:12,170 –> 00:08:13,609 Indeed this comes out of
319 00:08:13,619 –> 00:08:15,579 the root criterion for the.
320 00:08:15,730 –> 00:08:16,989 Therefore, we can calculate
321 00:08:17,000 –> 00:08:18,609 the lim sup of the cave
322 00:08:18,619 –> 00:08:20,309 root of the
323 00:08:20,320 –> 00:08:22,109 absolute value of A K.
324 00:08:22,609 –> 00:08:24,269 And this gives us the number
325 00:08:24,279 –> 00:08:25,549 one over R.
326 00:08:26,119 –> 00:08:27,549 And here in this case for
327 00:08:27,559 –> 00:08:29,230 zero and infinity, we
328 00:08:29,239 –> 00:08:31,089 use this symbolic calculation
329 00:08:31,589 –> 00:08:33,030 using this, we have this
330 00:08:33,039 –> 00:08:34,390 formula in general.
331 00:08:34,960 –> 00:08:35,308 OK.
332 00:08:35,320 –> 00:08:36,789 Now this nice formula is
333 00:08:36,799 –> 00:08:38,409 often called the Cauchy Hadamard
334 00:08:38,679 –> 00:08:39,058 theorem.
335 00:08:39,719 –> 00:08:41,130 With this, I would say we’ve
336 00:08:41,140 –> 00:08:42,558 learned a lot today and we
337 00:08:42,570 –> 00:08:44,140 can start with derivatives
338 00:08:44,150 –> 00:08:45,200 in the next video.
339 00:08:45,780 –> 00:08:46,900 Therefore, I hope I see you
340 00:08:46,909 –> 00:08:48,270 there and have a nice day.
341 00:08:48,400 –> 00:08:49,070 Bye.