Checking whether total variation of a function is a norm on BV

Question

Let's consider BV space which denotes set of functions with finite variation $f: \mathbb{R} \mapsto \mathbb{R}$.

I wonder whether (total variation) $V_{f}(\mathbb{R}) = \lim \limits_{x \to \infty} \sup \sum \limits_{i = 1}^N \lvert f(x_i) - f(x_{i-1}) \rvert$ is a norm on BV? The triangle inequality and multiplying is obvious.

Answer 1

If $V_f=0$ then $f$ is constant but not necessarily zero.