Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / E / 1 / a

A continuous map $f:X\to Y$ is one for which $f^{-1}(U)$ is open in $X$ whenever $U$ is open in $Y$. This defines continuity using the two topologies, without a metric or a coordinate description.

For metric spaces, continuity at $x_0$ means that for every $ϵ>0$ there is $\delta >0$ such thatThe choice of $\delta$ may depend on both $ϵ$ and $x_0$.