My question is: If $f_n\to f$ pointwise in $(0,1)$ where $f_n\in C^1$ (continuously differentiable), AND the family $\{f_n'\}$ is uniformly bounded and equicontinuous, then is $f\in C^1$ as well? Ie, is $f$ also continuously differentiable?
My intuition tells me that $|f_n'(x)|\leq M$ and equicontinuity should be enough to force $f'(x)$ to exist and be continuous but I'm not sure where to start a rigorous proof.
Any help is welcome.
Since differentiabilty is a local property we might as well work within a closed interval contained in $(0,1)$. Fix a point $c$ in this interval. We can apply Arzelà-Ascoli Theorem to find a subsequence $(f_{n_k})$ which converges uniformly to some continuous function $g$. Now $f_{n_k}(x)=f_{n_k}(c)+\int _c^{x} f_{n_k}'(t)dt$. Letting $k \to \infty$ we get $f(x)-f(c)=\int_c^{x} g(t)dt$. This implies that $f$ is continuously differentiable.