This is a two part question. I'm done the first part but a bit stumped with the second. Just posting the whole thing for clarity.
For each $n$, let $f_n: [0,1] \to \mathbb{R}$ be a continuous function which satisfies $f_n(0) = 0$ and $f_n$ is continuously differentiable on (0,1) with $|f'_n(x)| \leq x$ for $x \in (0,1)$.
- Prove that there exists a subsequence of $(f_n)$ which converges uniformly to a continuous function $f$.
- Must the limit function $f$ in Part 1 be differentiable on $(0,1)$? Prove or provide a counterexample.
The first part was straightforward to prove. I see some questions here on MSE that are related to part 2 and show certain sequences of continuous functions converging uniformly to a continuous function that is not differentiable. But it seems like I have a greater number of restrictions on my functions $f_n$ and I cant figure out whether these restrictions are enough to force $f$ to be differentiable on $(0,1)$.
Hint: start by finding an $f$ that is continuous, differentiable except at $1/2$, and satisfies $|f'(x)| \le x$ on $(0,1/2)$ and $(1/2, 1)$. Then find a sequence $f_n$ of differentiable approximations of $f$.