Bounded Variation imply L^1

Question

If a function $f:\mathbb{R}\to\mathbb{C}$ is of bounded variation, is it true that $f\in L^1(\mathbb{R})$?

We need to ask for $f$ to be continuous?

Answer 1

No this is false, for example $f(x)=x$ is a function of bounded variations but $\int _ \mathbb{R} |x|dx=+\infty$.