What is the most general solution of the functional equation $$f(x) = f(1/x)$$ for $x>0$?
Asked
Active
Viewed 82 times
1
-
4Take any function at all on the interval $(0,1]$. Then define it for larger $x$ via your functional equation. – lulu Oct 11 '16 at 21:50
1 Answers
2
Let $A$ be any set, let $g\colon(0,1]\to A$ be any function and define $f\colon (0,\infty)\to A$ by $$f(x)=\begin{cases}g(x)&\text{if }x\le 1\\g(\frac1x)&\text{if }x>1\end{cases} $$
Hagen von Eitzen
- 374,180
-
This way, $f$ is automatically continuous. What if we demand $f$ to be smooth? Do we get any restrictions? – Oct 11 '16 at 21:53
-
-
@LBO In general, my $f$ will not be continuous (after all, adding the condition of continuity would lessen the generality) – Hagen von Eitzen Oct 11 '16 at 22:22
-