Seeking examples of $f: (X,\tau) \rightarrow (Y, \tau) \;\text{continuous, but}\; f^{-1} (Y) \neq X$

Question

Could you give me some examples for this statement? $$f: (X,\tau) \rightarrow (Y, \tau) \;\text{continuous, but}\; f^{-1} (Y) \neq X$$

Answer 1

By definition $f^{-1}[Y] = \{x: f(x) \in Y \} = X$ as all points of $X$ map into $Y$ by $f$ being a map from $X$ to $Y$ by definition! Continuity of $f$ is irrelevant, this is just set theory and the definition of a function and of inverse images.