Till now, I have proved followings;
Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,
$Y=\mathbb{R}^k \Rightarrow \exists$ 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.
What are true and what are false if $f$ is replaced by a 'continuous function', not uniformly?