A continuos map preserves (path-)connectedness, compactness

Question

It is clear that continuous maps preserve those properties. What about the reverse? If a map maps every path-connected space to path connected then it is continuous? Also for connectedness, compactness And so on, what property are strong enough to indicate continuity?

Answer 1

1. (path) connected: No, take any discontinuous function $\mathbb{Q}\to\mathbb{Q}$. Note that $\mathbb{Q}$ is totally disconnected, i.e. a subset of $\mathbb{Q}$ is (path) connected if and only if it is a singleton.
2. compact: Again no, take any discontinuous function $X\to C$ where $C$ is finite discrete. Obviously image of every subset is compact.

what property are strong enough to indicate continuity?

It is unlikely you will find any such property for general topological spaces. In some specialized context it may be possible, e.g. being differentiable.