TITLE: Is f(x)=0 considered continous?
QUESTION [6 upvotes]: I'm aware that constant functions are considered continuous, but would $f(x)=0$ be considered a continuous function everywhere? Does $0$ count as a constant?

REPLY [8 votes]: Indeed it is continuous. Using the $\epsilon-\delta$ definition of continuity, we find that $\forall c\in \mathbb R$ and $\forall \epsilon > 0$ there exists $\delta > 0$ such that $\forall x\in\mathbb R$
$$|x-c| <\delta \implies |f(x) - f(c)| = 0 < \epsilon$$
Clearly, $\delta$ can be anything, so the continuity holds.

REPLY [6 votes]: Yes, it is continuous and 0 is a constant