If $f_n\to f$ uniformly and $f_n$ are differentiable, is $f$ differentiable?

Question

Suppose $f_n$ converges uniformly to $f$ and $f_n$ are differentiable. Is it true that f will be differentiable?

My initial guess is no because $f_n= \frac{\sin(nx)}{\sqrt n}.$ Is this right? And more examples would be greatly appreciated.

Answer 1

Your sequence converges uniformly to 0, which is differentiable. Consider $f_n(x)=\sqrt{x^2+1/n}$, which converges uniformly to $|x|$. Here is an outline of how you would prove that:

1. Prove that the largest difference between $f_n(x)$ and $|x|$ occurs when $x=0$.
2. Prove that if $n$ is large enough, that $|f_n(0)-|0||$ can be made arbitrarily small.
3. Explain why this means that the convergence is uniform.
4. Use the definition of the derivative to show that $|x|$ is not differentiable at $x=0$.

Answer 2

The limit function doesn't even have to be differentiable anywhere. Take any function $f: [0, 1] \to \mathbb{R}$ that is continuous but nowhere differentiable [such functions exist]. By the Stone-Weierstrass theorem you can find a sequence of smooth functions that converge uniformly to $f$ on $[0, 1]$.

For a more specific example, consider a Weierstrass function. The corresponding partial sums form a sequence of smooth functions that converge uniformly on $\mathbb{R}$ to a nowhere differentiable function.