Are $f(t)$ and $f'(t)$ bounded and piecewise continuous?

Question

I have the function

$$ f(t) = \left\{\begin{aligned} & -t , -2 < t \le 0\\ & t ,0 < t \le 2 \end{aligned} \right.$$

with periodicity of $4$ such that $f(t + 4) = f(t)$.

The derivative of this function is

$$ f'(t) = \left\{\begin{aligned} & -1 , -2 < t \le 0\\ & 1 ,0 < t \le 2 \end{aligned} \right.$$

Am I correct in saying that both $f(x)$ and $f'(x)$ are bounded and piecewise continuous?

Answer 1

Of course you are right : $f$ is bounded on $[-2, \ 2]$ and since it is a periodic function it is bounded everywhere $f'$ is also periodic by definition and therefore bounded for the same reason. $f$ is everywhere continuous and $f'$ is piecewise continuous.