Difference between a continuous function and an isometry? Is a continuous function a homomorphism? 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?
 A: No. $f$ is a isometry if
$$ d(x,y) = d(f(x),f(y))
$$
This is very different from the definition for continuity. You can consider the function $x \mapsto 2x$ on $\mathbb{R}$, it is continuous (choosing $\delta = \epsilon/2$) but not an isometry.
However, an isometry is necessarily continuous, since $f^{-1}[B(f(x);r)] = B(x;r)$. That is, choosing $\delta = \epsilon$.
A: Not quite actually.  A continuous function is one where points that are "close" in the domain are guaranteed to be within a certain distance from one another in the codomain.  Thus when we say two points are within $\delta>0$ of each other in the domain, we are guaranteed that under a continuous map their images will be within $\epsilon>0$ of each other.  The difference between continuity and an isometry is that an isometry will preserve the distance exactly.  That is, if $d_X$ is the metric on $X$ and $d_Y$ is the metric on $Y$, an isometry $f:X \to Y$ will satisfy
$$
d_X(p_1, p_2) \;\; =\;\; d_Y(f(p_1), f(p_2))
$$
which is not always guaranteed of continuous maps.
A: Constant functions are always continuous; they are never an isometry if $X$ has more than one point.
