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The Du Bois-Reymond theorem says that continuity of a periodic function does not guarantee convergence of its ordinary Fourier series at every point. At any prescribed circle point $x_0$, there is even a real continuous $f$ with unbounded Fourier partial sums $S_{N}f(x_0)$.

For $x_0=0$, view $T_N:f\mapsto S_{N}f(0)$ as bounded linear functionals on the real Banach space $C(\mathbb{T})$ with its supremum norm. The Dirichlet kernel harmonic lower bound givesIf every continuous input had bounded values under all $T_N$, the Uniform boundedness principle would bound these operator norms, a contradiction. Translation moves the exceptional point to any specified $x_0$. The generic unbounded Fourier sums at a fixed point strengthens this existence proof to a category statement.^[ref]

This does not contradict mean-square convergence: a continuous function is square-integrable, and its series still converges in $L^2$. Nor does it obstruct Fejér's theorem, which averages partial sums and gives uniform convergence for every continuous periodic input. The Fejér kernel is positive and its sums are a uniform-norm contraction, but the sums themselves need not be nonnegative: for $f=-1$, every Fejér sum equals $-1$.

The real space $C(\mathbb{T})$ is a Banach space in the uniform norm: a uniformly Cauchy sequence has a uniform pointwise limit by completeness of $\mathbb{R}$, and that uniform limit is continuous. Each functional $L_N$ from the preceding part is bounded, but their norms are unbounded.

For integers $m\ge 1$, defineEach $E_m$ is closed, being an intersection of inverse images of closed intervals under continuous functionals. It has empty interior. Otherwise some ball $B(f_0,r)$ would be contained in it. For every $h$ of norm at most one, both $f_0$ and $f_0+(r/2)h$ would belong to that ball, soTaking the supremum over $h$ would give $\Vert L_N\Vert \le 4m/r$ for all $N$, contradicting the preceding part.

Thus the $E_m$ are nowhere dense. The Baire category theorem says their union cannot cover $C(\mathbb{T})$. A function outside that union satisfiesso its Fourier series diverges at zero. In fact these functions form a dense $G_{\delta }$ subset of $C(\mathbb{T})$. This proves the relevant Uniform boundedness principle argument directly and establishes generic unbounded Fourier sums at a fixed point.