Exam (elaborations)
Math 302 Final Exam Questions And Answers Latest Update
- Course
- MATH 302
- Institution
- University Of Southern California
Math 302 Final Exam Questions And Answers Latest Update
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Solution 2024/2025Pepper
Math 302 Final Exam Questions And Answers
Latest Update
Local Continuity of a Function ANS✔✔ - Let (M, d) and (N, s) be metric
spaces.
- Let S be completely contained in M, with c in S
- Let f: S -> N be a function
We say f is continuous at c if the following holds:
limf(xn) = f(c) = f(limxn)
Global Continuity ANS✔✔ If f is continuous at all c, it is globally continuous
on S
- to prove, take function and prove continuity at general c, then state c is
arbitrary
Which functions from (M, discrete) -> (N, s) is continuous? ANS✔✔ All
What functions from R -> R are continuous? ANS✔✔ Polynomials, sinx, cosx,
e^x and their compositions, sums, difference, and product
f^-1 behaves nicely when it comes to intersections, unions, and
complements ANS✔✔ - f^-1(C /\ D) = f^-1(C)/\f^-1(D)
- f^-1(C U D) = f^-1(C) U f^-1(D)
- f^-1(C \ D) = f^-1(C) \ f^-1(D)
, Solution 2024/2025
Pepper
Why is R^n -> R, f(x) = x continuous? ANS✔✔ it is the projection of x onto
the first coordinate of x
Global Continuity Theorem ANS✔✔ TFAE:
(1) f is continuous on M
(2) Under f, the inverse image of an open set is open
(3) Under f, the inverse image of a closed set is closed
Corollary of Global Continuity Theorem ANS✔✔ Let f: (M, d) -> (R, usual) be
continuous
Let gamma = real valued number
(1) {x in M | f(x) < gamma} is open in M
(2) {x in M | f(x) > gamma} is open in M
(3) {x in M | f(x) <= gamma} is closed in M
(4) {x in M | f(x) >= gamma} is closed in M
(5) {x in M | f(x) = gamma} is closed in M
The continuous image of a ____ set is _____ ANS✔✔ compact **FORWARD
NOT INVERSE**
Max-Min Theorem ANS✔✔ A continuous real valued function on a compact
set contains its max and min in that set
Convex Definition ANS✔✔ A set C in a metric space is convex if the line
segment joining any two points in C is back in C
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