Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is continuous. Show that all the below are false,
$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.
$Y$ is compact $\Rightarrow \exists$ a continuous extension.
$Y$ is complete $\Rightarrow \exists$ a continuous extension. (AC$_\omega$)
$E$ is countable & $Y$ is complete $\Rightarrow \exists$ a continuous extension.
Note: If the function between spaces were uniformly continous, then such an extension would infact exist.