Proof:

Denote the adjoined points in $\widehat{X}$ and $\widehat{Y}$ as $p$ and $q$ respectively and extend the given map by sending $p$ to $q$. We need to show that the extension is continuous at $p$. Let $C$ be any compact subset of $Y$ so that $K = f^{-1}(C)$ is compact in $X$. Then $N = \widehat{X} - K$ is a neighborhood of $p$ in $\widehat{X}$ that is mapped by $\widehat{f}$ into the preassigned neighborhood $\widehat{Y} -C$ of $q$. This proves the continuity of the extension.

The converse is not true as the constant map shows. However the following version in the reverse direction is easy to see,