Let $f_n$ be a sequence of functions which converges to $f_1$ on $A$ and $f_2$ on $B,$ where $A$ and $B$ are disjoint subsets of $R^n$. I'm wondering wheather or not $f_n$ converges to $h$ on $A \cup B$?
$$ h\left(x\right) = \left\{ \begin{array}{l} {f_1}\left( x \right), \quad x \in A\\ {f_2}\left( x \right) , \quad x \in B \end{array} \right.$$
Could anyone help me to give a solution for the problem! If this is wrong, then can you provide a couter-example? thank you!