If f $\in$ $L^{1}[\pi]$ such that fourier coefficients of $f$ are absolutely summable. Then can we say that partial fourier sums of $f$ converge to $f$ pointwise almost everywhere?
In case it is true then can it be used to show that $f$ is continous almost everywhere?
Yes. If $\sum a_n e^{inx}$ is the Fourier series of $f$ then the series converges uniformly. Let the sum be $g$. Then $g$ is continuous and periodic. The Fourier coefficients of $g$ can be calculated by integrating term by term and we get $ \hat {g} (n) =a_n =\hat {f} (n)$. The fact that $f$ and $g$ are two $L^{1}$ functions with the same Fourier coefficients implies that $f=g$ almost everywhere. Hence the partial sums of the Fourier series converge to the continuous function $g$ at every point and $f=g$ almost everywhere.