Assume that $f$ is 2$\pi$ periodic and continuous on [-$\pi ,0)\cup (0, \pi$] with a jump discontinuity at x=0. Prove that the ceasro sums of the fourier series of $f$ converge to the halfway point of the jump continuity.
$\lim_{n\to \infty} \sigma_n (f)(0)=1/n\lim_{n\to\infty}\int_{-\pi}^\pi f(t)k_n(t)dt=1/2(f(0^-)+f(0^+))$
I have no idea how to start this problem
You don't need the continuity of $f$.
Proposition: If $f \in L^1(\Bbb{T})$ and $f(0^-)$ and $f(0^+)$ both exist then $$ \sigma_nf(0) \to \frac{f(0^-) + f(0^+)}{2}\quad(n\to\infty) $$
Proof: Let $$ g(t) := \begin{cases} \phantom{-}1-t/\pi & \phantom{-}0 < t \leq \pi \\ -1+t/\pi & -\pi \leq t < 0 \\ \phantom{-}\phantom{-}0 & \phantom{-}t = 0\end{cases} $$ and $$ \tilde{f}(t) := \begin{cases} f(t)-\frac{f(0^+)-f(0^-)}{2}g(t) & t \neq 0 \\ \phantom{-}\phantom{-}\phantom{-}\frac{f(0^+)-f(0^-)}{2} & t = 0 \end{cases} $$ Then $\tilde{f} \in L^1(\Bbb{T})$ and $\tilde{f}$ is continuous at $0$.
Note that $g$ is an odd function hence $\hat{g}(-k)=-\hat{g}(k)$ for all $k \in \Bbb{Z}$. In particular, $$ s_ng(0) = \sum_{k=-n}^n \hat{g}(k)=0 $$ so $$ \sigma_ng(0)=0\quad(n\geq0) $$ Hence \begin{align} \sigma_nf(0) &= \sigma_n\tilde{f}(0) \\ &\to \tilde{f}(0) & \text{by Fejér's theorem}\\ &=\frac{f(0^-) + f(0^+)}{2} \end{align} as $n\to\infty$.