Problem 533
Let \(\displaystyle f:\R \to \R\) be continuous and periodic. Does it imply that \(\displaystyle f(x)\) is bounded? Difficulty: 3. |
Problem 534
(Brouwer fixed point theorem; 1-dimensional case.) All \(\displaystyle f:[a,b]\to [a,b]\) continuous function has a fixed point, i.e., an \(\displaystyle x\), for which \(\displaystyle f(x)=x\). Difficulty: 3. Solution is available for this problem. |
Problem 535
Let \(\displaystyle f:[0,1] \to [0,1]\) and \(\displaystyle g:[0,1] \to [0,1]\) be continuous and \(\displaystyle f(0)\geq g(0)\), \(\displaystyle f(1)\leq g(1)\). Prove that there exists an \(\displaystyle x\in [0,1]\), such that \(\displaystyle f(x)=g(x)\). Difficulty: 3. |
Problem 537
Let \(\displaystyle f:[0,2]\to \R\) be continuous, \(\displaystyle f(0)=f(2)\). Prove that the graph of \(\displaystyle f\) has a chord of length 1. Difficulty: 4. |
Problem 542
Prove that every polynomial of odd degree has a real root. Difficulty: 4. |
Problem 543
Prove that the polynomial \(\displaystyle x^3-3x^2-x+2\) has 3 real roots. Difficulty: 4. Solution is available for this problem. |
Problem 547
Prove that if \(\displaystyle f:[a,b]\to\R\) is continuous and \(\displaystyle x_1,x_2,\ldots,x_n\in[a,b]\), then there is a \(\displaystyle c\in[a,b]\), for which \(\displaystyle f(c)=\dfrac{f(x_1)+\dots+f(x_n)}{n}\). Difficulty: 4. |
Problem 539
Prove that if \(\displaystyle I\) is an interval (closed or not, bounded or not, might be a point) and \(\displaystyle f:I\to\R\) is continuous, then \(\displaystyle f(I)\) is also an interval. Difficulty: 5. |
Problem 544
Prove that the continuous image of a compact set is compact. Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |