Problem 1299
\(\displaystyle f:\R^p\to \R^q\), \(\displaystyle A,B\subset \R^p\). Assume that \(\displaystyle f\) is continuous when restricted to either \(\displaystyle A\) or \(\displaystyle B\). Is it true that \(\displaystyle f\) is continuous when restricted to \(\displaystyle A\cup B\)? Difficulty: 3. |
Problem 1300
Let \(\displaystyle f:\R^p\to \R^q\), \(\displaystyle A,B\subset \R^p\) be closed. \(\displaystyle f\) is continuous when restricted to either \(\displaystyle A\) or \(\displaystyle B\). Is it true that \(\displaystyle f\) is continuous when restricted to \(\displaystyle A\cup B\)? Difficulty: 3. |
Problem 1298
\(\displaystyle f:\R^p\to \R^q\), \(\displaystyle A,B\subset \R^p\), \(\displaystyle x\in A\cap B\). 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? Difficulty: 5. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |