pow:problem6f23
Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revision | |||
| pow:problem6f23 [2023/11/21 08:12] – mazur | pow:problem6f23 [2023/11/21 08:13] (current) – mazur | ||
|---|---|---|---|
| Line 1: | Line 1: | ||
| + | <box 85% round orange| Problem 6 (due Monday, November 20)> | ||
| + | Find all bounded continuous functions $f:\mathbb R\longrightarrow \mathbb R$ which satisfy the following condition: | ||
| + | \[ f^2(x)-f^2(y)=f(x+y)f(x-y)\ \ \text{for all $x,y\in \mathbb R$}.\] | ||
| + | Here $f^2=f\cdot f$ is the square of the function $f$ (and not the composition of $f$ with itself). | ||
| + | </ | ||
| + | |||
| + | We received two attempts at solving the problem, both far from being correct. The functions which satisfy the conditions | ||
| + | of the problem are exactly all functions of the form $f(x)=A\sin bx$ for some real numbers $A,b$. For detailed | ||
| + | solutions (we have two) see the following link {{: | ||