1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ and proved it to be continuous but nowhere differentiable. Google finds some papers, but I am unable to find an easy proof for the function being continuous. (The fact that this function is nowhere differentiable seems to be more interesting to many papers …)
My own try: All partial sums are continuous as compositions of continuous functions, so I “only” have to show that the partial sums converge uniformly to $f$. But how to do that?