 Generic unbounded Fourier sums at a fixed point

For each integer $m\ge 1$, the set of continuous functions satisfying $|S_N(f,0)|\le m$ for every $N$ is closed in the uniform norm. If it contained a ball of radius $r$, subtracting two members of that ball would bound every partial-sum functional norm by $4m/r$. The Dirichlet kernel harmonic lower bound excludes this, so these sets are nowhere dense. By the Baire category theorem, their complement is a dense $G_{\delta }$ set. Every function in that complement has unbounded Fourier partial sums at zero. Translation gives the same conclusion at any prescribed point. This is a direct Uniform boundedness principle argument on the real Banach space of continuous functions on the circle.