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Continuous function

Definition

Given topological spaces $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$, a function $f: X \to Y$ is said to be continuous if the preimage of any open set is open, that is $U \in \mathcal{T}_Y \Rightarrow f^{-1}(U) \in \mathcal{T}_x$.

NOTE: this seems like a slightly backwards definition, but it makes a lot of sense when thinking about connectedness.

In metric spaces

We define the notion of continuity at a point $a$ in $X$. A function is continuous if, given a ball in $Y$ [about $f(a)$], we can fit a ball [about $a$] in X. TODO there's a subtlety here that is slightly wrong. the generalisation doesn't come from ignoring those bits in square brackets TODO

all these pages adapted with probably insufficient credit from my university's lecture notes