Prove that $\max(f(x),g(x)) = 0 \iff \min(-f(x), -g(x)) = 0$ ($f,g$ continuous) Let $f,g: \mathbb{R}^N \to \mathbb{R}$ be continuous functions. How can I prove that $$\max(f(x),g(x)) = 0 \iff  \min(-f(x), -g(x)) = 0.$$

I know that $\max(f) = -\min(-f)$. How can we deduce the result from this?

$$\max(f(x), g(x))=-\min(-f(x),-g(x))$$
Since $\max(f(x),g(x))=0$,
We have $-\min(-f(x),-g(x))=0$
and hence $\min(-f(x),-g(x))=0$.
The other direction is similar.
