Uniform limit of a sequence of bounded derivatives is a bounded derivative?

Question

Let $\{f_n\}$ be a sequence of differentiable functions on $\mathbb R$ such that $f_n'$ is bounded for each $n$ ; if $\{f_n'\}$ converges uniformly to $f$ on $\mathbb R$ then is it true that $f$ is bounded and the derivative of some function ?

Answer 1

Using $f$ as the limit of $f'_n$ is a little confusing, I will use $g$ as the limit function.

Let $h_n(x) = \int_0^x f'_n(t) dt$. Then $h_n(0) = 0$, and $h'_n = f'_n$.

Pick some interval $[a,b]$ that contains $0$. Using the result here Uniform convergence of derivatives, Tao 14.2.7. we see that $h_n$ converge uniformly (on $[a,b]$) to some limit $h$ such that $h' = g$.

Since the interval was arbitrary, we see that $h'=g$ everywhere.

Boundedness follows from uniform convergence to $g$ and the fact that each $f'_n$ is bounded.

Answer 2

$f$ is indeed bounded. By uniform convergence, there is some $N$ such that $\forall x\in \mathbb R, |f(x)-f'_N(x)|\leq 1$.

Then $\forall x\in \mathbb R, |f(x)|\leq 1+|f'_N(x)|\leq 1+M_N$