Problem 554
Are the following functions uniformly continuous? a) \(\displaystyle x^2\) on \(\displaystyle (1,2)\), b) \(\displaystyle \sin x\) on \(\displaystyle \R\), c) \(\displaystyle \sin{1\over x}\) on\(\displaystyle (0,\infty)\), d) \(\displaystyle 1/x\) on \(\displaystyle (0,2)\), e) \(\displaystyle \sqrt{x}\) on \(\displaystyle (0,\infty)\). Difficulty: 4. |
Problem 555
\(\displaystyle f,g:\R\to\R\) are uniformly continuous. Does it imply that \(\displaystyle f\cdot g\) is also uniformly continuous? Difficulty: 4. |
Problem 557
Prove that if \(\displaystyle f:\R\to\R\) is uniformly continuous on \(\displaystyle \R\) then the function \(\displaystyle f(x+5)-f(x)\) is bounded. Difficulty: 4. |
Problem 561
Let \(\displaystyle f:[0,1)\to\R\) be continuous. Prove that \(\displaystyle f\) is uniformly continuous if and only if \(\displaystyle \lim\limits_{1-}f\) exists and finite. Difficulty: 5. |
Problem 563
Let \(\displaystyle K\subset\R\). Prove that if all continuous \(\displaystyle K\to\R\) functions are uniformly continuous, then \(\displaystyle K\) is compact. Difficulty: 8. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |