Let X a Hausdorff Tychonoff space, C a closed subset of X, A a compact subset of int C and B a closed subset of C such that A and B are disjoint. Let f: A---> [0,1] and g: B---> [0,1] be continuous functions. Is it true that there exist a continuous extension h with domain C of both functions f and g? We know that f has such an extension, by compacity of A and X being Tychonoff, can we do that extension in a way to be also an extension of g? Thank you in advance.
No. Indeed, such an extension need not exist even if $A$ is empty, so you are just asking for an extension of $g$. For instance, let $X=C$ be any Tychonoff space that is not normal (say, the deleted Tychonoff plank) and let $S$ and $T$ be disjoint closed subsets of $X$ that cannot be separated by open sets. Then the function $g$ on $B=S\cup T$ that is $0$ on $S$ and $1$ on $T$ cannot be extended continuously to $C$.
(Of course, if $X$ is normal then such an extension does always exist by the Tietze extension theorem.)