MATH 3210-004 SPRING 2026 - Week 5 - Continuity 2 скачать в хорошем качестве

MATH 3210-004 SPRING 2026 - Week 5 - Continuity 2

--- This is MATH 3210-004, the advanced single variable calculus class at the University of Utah. View the complete course: https://github.com/AlpUzman/MATH_3210... --- Table of Contents: 00:00:00 1 of 3 00:00:17 midterm 1 00:02:07 recap: continuity at a point, continuity, C^0, C^0_b 00:07:46 little oh of 1 00:10:15 preview: Taylor approximation 00:13:40 blackboxing 00:18:13 exercise: equivalence in the abstract metric space case 00:19:26 proof of equivalence of two definitions of continuity at a point 00:36:13 farming a logical statement 00:37:05 discussion 00:45:28 homeomorphism, Homeo(X;Y), Homeo(X) 00:48:16 claim: composition of continuous functions is continuous 00:49:31 proof 00:50:00 exercise 00:50:21 2 of 3 00:52:35 exercise: algebraic properties of sets of real valued continuous functions 00:53:27 discussion of maximum principle 00:56:08 theorem: maximum principle (aka Extreme Value Theorem) Let f be a real valued continuous function on a closed and bounded interval. Then (i) f is bounded, (ii) the max and min of f exist 00:56:35 discussion continued 00:58:32 proof 01:24:28 exercise: any subsequence of a convergent sequence converges to the limit of the original sequence 01:25:14 proof continued 01:26:47 exercise: min part of theorem 01:27:21 discussion of theorem and proof 01:30:33 compact subset of metric space 01:31:16 maximum principle for X a compact metric space 01:32:35 discussion of intermediate value theorem 01:35:01 theorem: intermediate value theorem: Let f be a real valued continuous function defined on an interval. Then f attains any value between any two values it attains 01:37:26 proof 01:41:04 3 of 3 01:41:53 recap: IVT 01:42:15 heuristics for IVT 01:50:25 use of caricatures 01:52:33 proof of IVT 01:53:04 exercise: farm a sequence that converges to sup 01:53:42 proof continued 02:02:44 discussion of proof 02:05:37 theorem: Let I be a closed and bounded interval, f be a real valued function on I. Then (i) if f is C^0, then its image is [min(f),max(f)]. (ii) if im(f) is a closed and bounded interval and f is strictly monotone, then f is C^0 (iii) if f is C^0 and strictly monotone, then f is a homeo 02:10:50 recap: homeo and strict monotonicity 02:12:25 discussion of theorem 02:13:58 proof of theorem part (i) 02:17:53 discussion of proof 02:18:18 proof of theorem part (iii), assuming part (ii) 02:20:24 exercise 02:21:30 proof continued 02:22:30 heuristics for proof of theorem part (ii) --- License: CC BY-NC-SA 4.0 Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License https://creativecommons.org/licenses/... Alp Uzman https://alpuzman.github.io/