So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In general, $S_Nf$ does not converge to $f$ in $L^\infty(\mathbb{T})$ for continuous $f$ correct? ($\mathbb{T}$ denote the one dimensional torus $[0,1)$.)
Many thanks