[Math] Piecewise $\mathscr C^1$ and piecewise continuous

Question

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function and find its sine Fourier series. Can we say that this function is then piecewise continuous, but not piecewise $\mathscr C^1$? Or is it piecewise $\mathscr C^1$?

Would a constant odd function like $f(x)=C$ be piecewise $\mathscr C^1$?

Thanks for helping me clear my gaps.

Answer 1

This function is piecewise continuous on $\mathbb R$, and it is $\mathscr C^1$ on the continuous parts.

Therefore we can say it is piecewise $\mathscr C^1$ on $\mathbb R$