Department of Mathematics

Colorado State University

Math 317: Homework 9

Due: Friday, November 15, 2019

Reading: Read sections (30,) 31, 32

Exercises:

  1. Let when is rational and otherwise.

    1. Show that is continuous at .

    2. Show that is discontinuous at all .

    3. Prove that is differentiable at . (It is insufficient to simply claim )

  2. Suppose is differentiable at .

    1. Prove that .

    2. Prove that .

  3. Let \(f(x) = x \sin \frac 1x\) for \(x \ne 0\) and \(f(x) = 0\) for \(x = 0\). Note that \(f\) is continuous on \(\mathbb R\) (in fact, it is uniformly continuous on \([-R, R]\) for any \(R \ge 0\)).

    Prove that \(f(x)\) is not differentiable at \(x = 0\).