In my topology class, we have proved that if $f$ is a continuous function between two topological spaces, then $f$ preserves connectedness and it preserves compactness (assuming the domain is connected and/or compact).
We have also seen that $f$ preserves paths between two spaces.
What other structure does a continuous map preserve?
Is there some structure that a continuous map does not preserve?
Suppose $f$ is a continuous map from the separable topological space $X$ to the topological space $Y$ and $A$ is a countable dense subset of $X$.
Let $V$ denote an open set in $Y$ with inverse image the open set $U$ in $X$. Let $a\in A\cap U$. Then $f(a)\in V$. Thus $f(A)$ is a countable dense subset of $Y$, so $Y$ is separable.