A continuous nowhere differentiable function

Counterexample

Let $\varphi \colon \mathbb{R} \to \mathbb{R}$ be the zigzag function with $\varphi(x) = |x|$ for $-1 \le x \le 1$ and $\varphi(x + 2) = \varphi(x)$ for all $x$. Define $$f(x) = \sum_{n=0}^{\infty} \Big( \frac{3}{4} \Big)^n \varphi(4^n x).$$ The function $f$ is continuous on $\mathbb{R}$, but it is not differentiable at any point.

Remarks

Since $0 \le \varphi \le 1$, the M-test with $M_n = (3/4)^n$ and the continuity of uniform sums give continuity. Fix $x$. For each $m$, choose $\delta_m = \pm \frac{1}{2} 4^{-m}$ so that no integer lies strictly between $4^m x$ and $4^m (x + \delta_m)$. Then $\Big| \frac{f(x + \delta_m) - f(x)}{\delta_m} \Big| \ge \frac{3^m + 1}{2}$. This tends to infinity while $\delta_m \to 0$, so $f'(x)$ does not exist. This function replaces the cosine in the Weierstrass function by a piecewise linear zigzag.

Explanations

L1 human

Each term adds smaller and steeper zigzags to the graph. The sum is continuous, but its graph has a corner at every scale near every point.