Extension theorems in a topological space

Question

I have to prove some cases of extention problems one of them is : If $X$ be a completely regular space and $E$ is a subspace of $X$ which consists of a closed set $F$ of $X$ and a point $p \in X \setminus F$, if $g:E\to I$ denotes the continuous map defined by $$g(x)=\begin{cases} 0 &\text{ if } x=p\\ 1 & \text{ if } x \in F\end{cases}$$ then $g$ has an extension $f:X\to I$