Exercise 2, § 1

On the interval $I = [0, 1]$ we define a sequence of continuous real functions $(f_n)$ inductively as follows: We take $f_0(x) = x$; for each integer $n \geq 1$ the function $f_n$ is affine linear on each of the $3^n$ intervals $\left[ \frac{k}{3^n}, \frac{k+1}{3^n} \right]$ for $0 \leq k \leq 3^n - 1$; further, we take

Show that the sequence $(f_n)$ converges uniformly on $I$ to a continuous function which has no derivative (finite or infinite) at any point of the interval $]0, 1[$ (use exerc. 1).