I have the function
$$ f(t) = \begin{cases} \dfrac{1}{2} + t & \text{when, } -\dfrac{1}{2} < t \le 0,\\ \dfrac{1}{2} - t & \text{when, } 0 < t \le \dfrac{1}{2}. \end{cases} $$
and its periodic extension, $f(t + 1) = f(t)$.
I calculated that $\lim_{t \to 1/2^+} f(t) = \lim_{t \to -1/2^+} f(t + 1) = 1 \not = \lim_{t \to 1/2^-} = 0$.
We know that, if $f(t)$ and $f'(t)$ are bounded and piecewise continuous on $[-L, L]$, then the Fourier series converges to $f(t)$ except at points of discontinuity, where it instead converges to $\dfrac{f(t^+) + f(t^-)}{2}$ where $f(t^+)$ is the right-hand limit and $f(t^-)$ is the left-hand limit.
Therefore, would I be correct in saying that the Fourier series of this function DOES NOT converge to $f(t)$ for all $t$?
I would greatly appreciate it if people could please take the time to review my work.