The definition of continuous function in topology I am self-studying general topology, and I am curious about the definition of the continuous function. I know that the definition derives from calculus, but why do we define it like that?I mean what kind of property we want to preserve through continuous function?

The reason is that $\delta$-neighbourhood and $\epsilon$-neighbourhood in the definition of the continuity of $f$ in Euclidean space respectively correspond to $U$ and $f^{-1}(U)$ in the topological sense. (I'm not explaining the details since you are just wondering the reason not the definition of all these)
