Why "preimage" in the definition of continuity?

Question: Why do we have "preimage" (of open or closed set) instead of "image" in the topological definition of continuous function?

Answer: This is a meta-question. The meta-answer is: because "for any $\epsilon$ there is a $\delta$" and not vice versa. So, you have a function $y=f(x)$ and start with $y$ and then go to find its counterpart on the $x$-axis. That's preimage!

Another interpretation is that this issue is about accuracy: achieve the required accuracy of the measurement of $y$ via that of $x$ (such as the area in terms of the radius of a circle).

Exercise: Find examples of continuous functions that don't preserve openness of sets, closedness of sets, take open to closed, and vice versa.

Notice that such properties as connectedness, compactness, etc are preserved under continuous functions.