Definition of continuous function on a set: Suppose $X$ and $Y$ are metric spaces. Let $f: X \to Y$. We say $f$ is continuous on $X$ if for every $\varepsilon >0$, $\exists \delta >0$ such that $d(x, x_0) < \delta \implies d\left(f(x), f(x_0)\right) < \varepsilon$, $\forall x \in X$.
I learned, not too in depth, that an isometry is a map which preserves distance.
Isn't that what is going on with the definition of a continuous function? What is the difference?
Also, it seems like a continuous function is basically a homomorphism between metric spaces that preserves some sort of distance. Is this correct?