The proof of the existence of a continuous function $f\in C(\Bbb T)$ with divergent Fourier series using the Banach-Steinhaus theorem is well-known. It uses the unboundedness of the Dirichlet kernel $D_n$ - note that $\|D_n\|_1 \sim \log n$. For example, see this answer.
I'm trying to extend the argument to a countable set of points. Let $\{x_k\}_{k=1}^\infty$ be a countable subset of $\Bbb T = \Bbb R/\Bbb Z$. Define the functionals $$\Lambda_{N,k}(f) = S_N(f;x_k) = \sum_{|n| \le N} \widehat{f}(n) e^{2\pi in x_k}$$ Then, by the Banach-Steinhaus theorem, if $$\sup_{N,k} \|\Lambda_{N,k}\| = \infty,$$ then there exists $f\in C(\Bbb T)$ such that $$\sup_{N,k} |\Lambda_{N,k} (f)| = \infty.$$ As the supremum is over $N$ and $k$, this doesn't seem to help us show that a function $f\in C(\Bbb T)$ exists such that its Fourier series diverges at every $x_k$, i.e., for every $k\in \Bbb N$, $$\sup_{N} |\Lambda_{N,k}(f)| = \infty.$$ Is there a way to fix this argument, i.e., use the Banach-Steinhaus theorem to arrive at the result? That's what I'd be most interested in. If not, it'd be nice to see other ways to show that such an $f$ exists. Thank you!