Say I have a function
$$f: \Pi \rightarrow \Phi$$
where $\Pi \subset \mathbb{R^2}, \Phi \subset \mathbb{R}$
If I were to say that $f \leq 0$ on $\partial\Pi$, then can I say
- $M \leq 0$ OR
- $M = 0$
Where $M$ is the maximum of $f$ on the boundary.
What I am getting at is that $f$ is less than OR equal to 0 on the boundary therefore it's maximum could be zero or possibly less.
This seems like an incredibly stupid question that I should know the answer to.