Continuous extension
= Continuous extension
A continuous extension of a <function> $f:D\to Y$ to a larger <domain> $E\supseteq D$ is a <continuous function> $F:E\to Y$ satisfying $F(x)=f(x)$ for $x\in D$. The <domain> and <codomain> carry specified <topologies>. Prescribing values on a dense <subset> does not by itself prove the existence of a continuous extension.