Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and let $D$ be the set of points where $f$ is differentiable. In order to show that $D\in\mathbf{\Pi_3^0}(\mathbb{R})$, I want to show that $x\in D$ if and only if for all $ n\in\mathbb{N}$ there exists $m\in\mathbb{N}$ such that $\left|\frac{f(x+h_1)-f(x)}{h_1}-\frac{f(x+h_2)-f(x)}{h_2}\right|\leq\varepsilon$ for all $h_1,h_2\in\mathbb{Q}\cap(-\frac{1}{m},\frac{1}{m})\setminus\{0\}$.
I have already shown that if $x\in D$, then $x$ satisfies the last condition and I'm currently trying to prove the other implication as follows:
If $x\notin D$, then for all $L\in\mathbb{R}$ there exists some $\varepsilon>0$ such that for all $\delta>0$ there is some $h\in(-\delta, \delta)\setminus\{0\}$ such that $\left|\frac{f(x+h)-f(x)}{h}-L\right|>\varepsilon$. Choose $n\in\mathbb{N}$ such that $\frac{1}{n}<\varepsilon$. Then, for all $m\in\mathbb{N}$ there is some $h\in(-\frac{1}{m},\frac{1}{m})\setminus\{0\}$ such that $\left|\frac{f(x+h)-f(x)}{h}-L\right|>\frac{1}{n}$. Given that $f$ is continuous and $\mathbb{Q}$ is dense in $\mathbb{R}$, there is some $h_1\in\mathbb{Q}\cap(-\frac{1}{m},\frac{1}{m})\setminus\{0\}$ such that $\left|\frac{f(x+h_1)-f(x)}{h_1}-L\right|>\frac{1}{2n}$. I think I'm missing a detail to finish this proof, but I haven't been able to find it.
Any help will be highly appreciated.
