Difference between convergence of Fourier Series with relaxed condition and main condition

Question

In the main theorem of Fourier Series, it converge to the mean of $f$ around $x$,which is $\frac{1}{2}\left[f(x+0)+f(x-0)\right]$,with condition that both $f$, $f'$ and $f''$ are sectionally continuous。 And in its corollaries, Fourier series converge to $f(x)$ with relaxed condition that is only $f$ and $f'$ are sectionally continuous. What's the difference? And are there any applications of the difference?