Set 8

  1. Rudin Chapter 5 #2
  2. Rudin Chapter 5 #11
  3. Rudin Chapter 5 #15
  4. Rudin Chapter 5 #16
  5. problems X213 (you may assume that Y is R^k)
  6. 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.
  7. 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)
  8. 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.
  9. 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.