Is the set of rational number discrete or continuous?

This does depend on the topology that we equip $\mathbf{Q}$ with. If it has its usual topology, i.e. the topology inherited from the standard topology on $\mathbf{R}$, then we have that it is not discrete. A topological space $X$ is said to be discrete if given any $x\in X$ there exists an open set $U$ containing $x$ such that $U\cap X=\{x\}$. Given any $\frac{p}{q}\in \mathbf{Q}$, and an open neighborhood of radius $\epsilon$, we can find another rational $\frac{m}{n}$ satisfying $\lvert \frac{p}{q}-\frac{m}{n}\rvert<\epsilon$, so that $\mathbf{Q}$ is not discrete.

To say that the reals are continuous does not mean anything. There is no such notion, strictly speaking. Functions are continuous. One could say that the reals are complete. The rationals then, although dense, are not complete.