We have the standard topological definition of continuity as:
For topological spaces, $(X,\tau_x)$ and $(Y,\tau_y)$, a function $f : X \rightarrow Y$ is continuous if $\forall V \in \tau_Y$, $f^{-1}(V) \in \tau_X$.
I was presented with a similar property, namely:
Let $f : X \rightarrow Y$, where $\forall U \subseteq X$, $f(U) \in \tau_Y$ implies $U \in \tau_X$
How is this different from the definition of continuity? It seems to me that the property above starts from the domain into the range, while continuity is the reverse. Perhaps, could anyone show me examples of $f$ that is continuous but the property does not hold, and vice versa?