I would like some clarification on exactly what a 'continuous invariant' of topological spaces is. My book does not give a straight definition but rather just says "Properties preserved by continuous functions are called continuous invariants." It motivates this with the theorem that states "Let $X$ be a connected space and $f:X \to Y$ a continuous function from $X$ onto a space $Y$. Then $Y$ is connected."
In this case, the 'onto' part of the above theorem is necessary. For example, take a map $f:\mathbb{R} \to X : x \mapsto a$ where $a$ is a fixed point in $X$ and $X$ is a discrete space with at least 2 points. Then this constant function is continuous but $X$ is not connected, since no discrete space with at least two points is.
My question is: Is the 'onto' part of the theorem above part of the general definition of a continuous invariant? Does a continuous invariant carry over to the codomain of a continuous map that isn't surjective, or only to its image? What is the (or a) formal definition for "continuous invariant"?