Given a sequence of twice differentiable functions $f_{n}: E → R$ where $E$ is an interval, suppose that $f_{n} → f$ pointwise to $E$, that $(f′_{n} (x_{0}))$ is bounded for some $x_{0}∈E$ and that $(f′′_{n})$ is uniformly bounded on $E$. Show that $f∈C^{1}$.
I have a idea to show this, first I want to use the theorem of continuity and uniformly convergence to show that sequence $f'$ is continuos , but I need to show that $(f_{n}')\rightarrow f'$ uniformly and I was looking in theorem of differentiable and uniformly convergence to do this and but again I need to show that sequence $(f′′_{n})\rightarrow f′′$ uniformly and I don't how I do after. If you can give me some advice ,I will be grateful. Thank you.
I suppose you are considering a compact interval $[a,b]$. Let $c$ be any piont in $[a,b]$.
$(f_n')$ is equi-continuous since by MVT. By Arzela-Ascoli theorem there is a subsequence $(f_{n_i})$ converging uniformly to a continuous function $g$. Now $f(x)=\lim f_{n_i}(x)=\lim f_{n_i}(c)+\lim \int_c^{x} f_{n_i}' (t) dt=f(c)+\int_c^{x} g(t)dt$. Since $g$ is continuous it follows that $f$ is continuously differentiable.