Question:
Give an example of a sequence of continuously differentiable function $(f_n)$ on $[0,1]$ so that $f_n \to f$ uniformly, but $f$ is not differentiable at all points of $[0,1]$.
My Thoughts:
Would a Fourier Series be a correct answer to this question? Take the Triangle Wave for instance. Wikipedia gives me the following equation:
Here $\omega$ is the angular frequency. Instead of $\infty$ in the sum, could each of my $f_n$ be $\sum_{k=0}^n$. In the limit, this sum of continuously differentiable functions converges to a function that is not differentiable at its cusps.