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.


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.


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