Convergent Fourier series of continuous function Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether one can find a continuous function $f$ with the property that its Fourier series is convergent everywhere but not to $f$ (in other words for each $x$ the partial sums $S_N(x)$ converge to $S(x)$ and there are some points $x$ such that $S_N(x)$ converges to $S(x)\neq f(x)$).
 A: If the partial sums $S_N(x)$ converge to $S(x)$, then the Cesaro sums
$$
\sigma_N(x)=\frac1N\sum_{k=1}^NS_N(x)
$$
also converge to $S(x)$. But the Cesaro sums of a continuous function $f$ converge uniformly to $f$, so that $S(x)=f(x)$ for all $x$.
A: The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s, resolved positively in 1966 by Lennart Carleson in $L^2$ and generalized by Richard Hunt to $L^p$ for any $p > 1$. This result is known as the Carleson–Hunt theorem.
This article may help.
Andrey Kolmogorov, as a student at the age of 19, constructed an example of a function in $L^1$ whose Fourier series diverges almost everywhere (Original Article)
A: @JuliánAguirre 's answer is correct.
But that's true only if the partial sum $S_N(x)$ converges at every $x$.
I guess what OP wanted to ask was if one could find a function $f$ whose Fourier series $S(x)$ doesn't converge to $f$ at some point. Note that this needs not to mean that $S(x)$ converges at every point. If $S(x)$ diverges at some point, then it clearly doesn't converge to $f$.
So I want to post a useful and relevant fact here.
Fact: There exist a continuous function $f$ whose Fourier series doesn't converge to $f$ on a null set (i.e. set of zero measure).
Here is a constructive proof from an anonym, c.f. Example of a function whose Fourier Series fails to converge at One point.
And here is an abstract proof from Wikipedia, c.f. https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Pointwise_convergence.
