Difference isometrically isomorphism and homeomophism. What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?

An isometry is a map $f:X\to Y$ between metric spaces that preserves distances: $d_Y(fx, fy) = d_X(x, y)$. Such maps are automatically continuous (just use the $\delta$-$\epsilon$ definition of continuity) and injective, but they may not be surjective; an isometric isomorphism is one that's both a bijection and an isometry. The inverse $f^{-1}$ is then also an isometry and thus continuous, so $f$ is also a homeomorphism. Homeomorphisms aren't necessarily isometries; take $x\to x^3$ on $\mathbb{R}$ with the usual metric.
