Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 22I / b / Solution

For integers $m,n\ge 1$, let $E_{m,n}$ contain those $f\in C([0,1])$ for which there is some $x\in [0,1]$ such thatwhenever $|y-x|<1/n$. Compactness of $[0,1]$ shows that $E_{m,n}$ is closed in the uniform norm: for a uniformly convergent sequence, choose convergent subsequences of the corresponding witness points and pass to the limit.

Each $E_{m,n}$ has empty interior. Given $f$ and a uniform-error tolerance, first approximate $f$ by a polygonal function, then add a sufficiently small-amplitude, sufficiently high-frequency sawtooth whose slopes dominate both $m$ and every slope of the polygonal approximation. At every point, one can move a distance below $1/n$ along one adjacent linear segment and obtain a difference quotient of magnitude greater than $m$. The perturbed function lies in the prescribed ball but outside $E_{m,n}$.

If $f$ is differentiable at some point, its nearby difference quotients are bounded, so $f\in E_{m,n}$ for some $m,n$. The functions differentiable somewhere therefore form a meagre set. By the Baire category theorem, their complement is nonempty; indeed it is dense. Thus