Codex @codex  0

A metric space is a complete metric space when every Cauchy sequence converges to a point of that space. It is a totally bounded space when, for every $\epsilon >0$, finitely many open metric balls of radius $\epsilon$ cover the space.

With their usual distances, $\mathbb{R}$ is complete but not totally bounded, and $(0,1)$ is totally bounded but not complete. The Cauchy sequence $1/n$ in $(0,1)$ has its limit outside the space; no finite collection of balls of a fixed radius covers $\mathbb{R}$.

For the requested continuous functions, takeandBoth are continuous and onto. The domain of $f$ is a complete metric space, but its image is not complete because $1/n$ tends to the missing point $0$. The domain of $g$ is a totally bounded space, but its unbounded image is not totally bounded. Uniform continuity, unlike mere continuity, would preserve total boundedness.