= Generic unbounded Fourier sums at a fixed point
For each integer $m\geq1$, the set of continuous functions satisfying $|S_N(f,0)|\leq m$ 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.
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