# Harrison Chapman

## Math 317: Homework 9

Due: Friday, November 15, 2019

Exercises:

1. Let $f(x) = x^2$ when $x$ is rational and $f(x) = 0$ otherwise.

1. Show that $f$ is continuous at $0$.

2. Show that $f$ is discontinuous at all $x \ne 0$.

3. Prove that $f$ is differentiable at $0$. (It is insufficient to simply claim $f'(x) = 2x$)

2. Suppose $f$ is differentiable at $a$.

1. Prove that $\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = f'(a)$.

2. Prove that $\lim_{h\to 0}\frac{f(a+h) - f(a-h)}{2h} = f'(a)$.

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 $$x = 0$$.