Q:

Differentiability at 0 for modified Dirichlet Function

Consider the modified Dirichlet function $f$ where $f(x) = x$ if $x$ is rational and $f(x) = 0$ otherwise. I have managed to prove that $f$ is continuous at and only at $0$.
My question is, then, "Is $f$ differentiable at $0$?" I have had a hard time proving this one way or the other using the following definition of the derivative at a point $c$:
$\lim\limits_{x \to c} \frac{f(x)-f(c)}{x-c}$.
I'd appreciate any help.

A:

We have that $$\lim_{x\to 0,x\in \mathbb{Q}}\dfrac{f(x)-f(0)}{x-0}=1$$ and
$$\lim_{x\to 0,x\in \mathbb{R}\setminus\mathbb{Q}}\dfrac{f(x)-f(0)}{x-0}=0.$$ Thus, $f$ has no derivative at zero.

