Continuous function with divergent Fourier series

Counterexample

There is a continuous $2\pi$-periodic function $f \colon \R \to \C$ whose Fourier series diverges at $0$; in fact $\sup_N \abs{S_N f(0)} = \infty$.

Check: let $C(\mathbb{T})$ be the Banach space of continuous $2\pi$-periodic functions with the supremum norm. For each $N$, the map $\Lambda_N(f) = S_N f(0) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) D_N(t) \, dt$ is a bounded linear functional on $C(\mathbb{T})$, where $D_N$ is the Dirichlet kernel. Its norm is the Lebesgue constant $L_N = \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{D_N}$, and $L_N \to \infty$. If $\sup_N \abs{\Lambda_N(f)}$ were finite for every $f \in C(\mathbb{T})$, the uniform boundedness principle would give $\sup_N L_N < \infty$, which is false. So some $f \in C(\mathbb{T})$ has $\sup_N \abs{S_N f(0)} = \infty$, and its Fourier series diverges at $0$.