Continuous function

In mathematics, a function f from a topological space $(X,O_{X})$ to another topological space $(Y,O_{Y})$, usually written as $f:(X,O_{X})\rightarrow (Y,O_{Y})$, is a continuous function if for every point $y\in Y$ such that $y=f(x)$ for some $x\in X$, and for every open set $U_{y}\in O_{y}$ containing y, there exists an open set $U_{x}\in O_{X}$ containing x such that $f(U_{x})\subset U_{y}$. Here $f(U_{x})=\{f(x')\in Y\mid x'\in U_{x}\}$.