Continuous maps in topology; the definition?

Yes, that is correct.

A function that maps open sets to open sets is called an open map, i.e a function $f : X \rightarrow Y$ is open if for any open set $U$ in $X$, the image $f(U)$ is open in $Y$.

Open maps are not necessarily continuous.

Then there is the concept of closed maps which maps closed sets to closed sets. A map may be open, closed, both, or neither and continuity is independent of openness and closedness.

A continuous function may have one, both, or neither property.

Continuous maps don't have to map open sets to open sets. An example is the map $f:\mathbb{R}\rightarrow\mathbb{R}$ given by $f(x)=x^2$ which maps $(-1,1)$ to $[0,1)$ which is not open and not closed.

Yes it is. Consider, for example, the continuous function $$f(x) = x^2$$ What is the image of the open set $(-1,1)$ ?