Final Missed Questions

Question 1

Let K be a compact set, and D be a closed set. Then K ∩ Dc is compact.

Answer 1

False

Question 2

Let (an) be a bounded sequence. Define bn = sup{ak : k >= n}. Then (bn) converges.

Answer 2

True

Question 3

Suppose (fn) is a sequence of functions defined on an open interval A where (fn) converges to a function f on A. If f is continuous, then the convergence is uniform.

Answer 3

False

Question 4

Suppose f is a bounded function on [0, 1] that is discontinuous
at a countable number of points. Assume the set of discontinuities, Df on [0, 1] are isolated (there are no limit points in Df , though there could be limit points of Df). Then, f is integrable.

Answer 4

True

Question 5

How to show that a function is continuous

Answer 5

Find a delta such that
|x-c| < d => |f(x)-f(c)| < ϵ

Question 6

How to show that a series is continuous

Answer 6

Usually Wierstrass M-test shows that if the function is less than another convergent function, then the series converges uniformly and is thus continuous

Question 7

How to manually show integrability criterion

Answer 7

Manually make a partition such that all the segments which have inf =/= sup add up to less than epsilon, thus U(f,P)-L(f,P) < ϵ