Can this fuctional equation be solved?

Question

$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$

$f(x^2)=xf(x)$

$f(x+1)=f(x)+1$

Can this functional equation be solved ?

Yes, it can be solved by you. – Watson Jun 12 '18 at 14:18 — Watson, Jun 12 '18 at 14:18
Or at the very least, attempted by you. – Rhys Hughes Jun 12 '18 at 14:21 — Rhys Hughes, Jun 12 '18 at 14:21
For $f$ continuous this has been solved on MSE here. – Dietrich Burde Jun 12 '18 at 14:34 — Dietrich Burde, Jun 12 '18 at 14:34

score 1 · Answer 1 · answered Jun 12 '18 at 14:22

1

Perhaps obvious but what about $$f(x)=x?$$

answered Jun 12 '18 at 14:22

Rhys Hughes

1

What about it? Do you think this is an answer? – nonuser Jun 12 '18 at 14:25
1

Yes. It works but just seemed too obvious. – Rhys Hughes Jun 12 '18 at 14:27
1

The point is to find, with proof, all solutions. No? – the_fox Jun 12 '18 at 14:29
More often than not, solving functional equations involves finding all solutions, not just one. So yes, it's probably too obvious. – Arthur Jun 12 '18 at 14:29

1 Answers1