Periodic function such that integral of $|x(t)|$ is finite

Question

I'm wondering if there is a periodic function such that the integral of $|x(t)|$ is finite. I tried with various functions, but the result is always infinite.

Thank you in advance.

Answer 1

$\int |x(t)|\, dt= \sum\limits_{n=-\infty}^{\infty} \int_{nT}^{(n+1)T} |x(t)|\, dt$. Use a change of variable to show that all the terms are equal to $\int_0^{T} |x(t)|\, dt$. Can you now conclude that $|x|$ is integrable iff it is $0$ almost everywhere?

Answer 2

Hint: suppose that $x$ has period $T$. Then what would the integral of $|x|$ look like in terms of its integral over one period?

Answer 3

$f(x)=0$ has period $T$ for all $T\in\mathbb{R}_{>0}$.