Convergence of Fourier series when $f(x)$ is defined on mixed interval

Question

I was wondering if you have a periodic, piecewise $C^1$ continuous function $f(x)$ defined on a mixed interval, e.g. $[-1, 1)$, if the $FS f(x)$ still converges uniformly on $[-1, 1)$? You know that $f(-1) = f(1)$.

Task

Theorem

Answer 1

I suspect the answer to the Question is no.

But you should ask about the question you really care about! The Fourier series for the triangle function $f$ in the "task" you added to a later version does converge uniformly on the line.

(In fact that function is piecewise $C^1$ on $[-1,1]$, making it very unclear where the question about $[-1,1)$ came from...)