Problem 1298 (difficulty: 5/10)

\(\displaystyle f:\R^p\to \R^q\), \(\displaystyle A,B\subset \R^p\), \(\displaystyle x\in A\cap B\). Assume that \(\displaystyle f\) is continuous at \(\displaystyle x\) when restricted to either \(\displaystyle A\) or \(\displaystyle B\). Prove that \(\displaystyle f\) is continuous at \(\displaystyle x\) when restricted to \(\displaystyle A\cup B\). Does this remain true for a union of infinitely many sets?

Give me another random problem!