- Rudin Chapter 5 #2
- Rudin Chapter 5 #11
- Rudin Chapter 5 #15
- Rudin Chapter 5 #16
- problems X213 (you may assume that Y is R^k)
- Suppose that f is a bijecction from R to R, where R denotes the real line, and that f is continuous. Prove that f(Q) is dense, where Q denotes the set of rationals.
- Suppose that f is differentiable on [a,b], for a< b real numbers, and that the absolute value of the derivative of f is bounded by M on [a,b]. Prove that, for each x and y in [a,b], |f(x) – f(y)| is bounded by M(b-a). (you can use the MVT)
- For each natural number n, compute the derivative of the function defined by x^n*sin(1/x) for x not zero and equal to 0 at zero AND determine if the derivative of x^n*sin(1/x) is continuous at zero.
- Review: Assume that f is a continuous function from a compact metric space X to a metric space Y.
(a) Prove that f(X) is compact.
(b) Need f(X) be closed and bounded? Why or why not?
(c) Assume further that f is one-to-one and onto. Prove f^{-1} is continuous.