# Is there a function such that $f(x+y) = f(xy)$? [closed]

If we define $$x, y \in \mathbb{R}$$, is there a function that fulfills the condition:

$$f(x+y) = f(xy)$$

for all $$x$$ and $$y$$?

For now, let's assume there are no stipulations on continuity and differentiability.

• Any constant function works. Aug 29, 2019 at 19:24
• Constant functions will satisfy this property. Aug 29, 2019 at 19:24
• Taking $y=0$ we see that $f(x)=f(0)$ identically.
– lulu
Aug 29, 2019 at 19:25
• Only constant functions would work, because $f(x+0)=f(0)$ for all $x$ Aug 29, 2019 at 19:25
• Have you read the comments? Several people have shown that only constants work.
– lulu
Aug 29, 2019 at 19:28

If $$y=0$$ then $$f(x)=f(x+0)=f(0x)=f(0)$$, so $$f$$ is a constant.
• @NoChance The title question hardly needs an answer, because $f=0$ is certainly such a function. So it makes sense to give an answer which takes into account the comments, too. Aug 29, 2019 at 20:24
• The problem here is that the domain of $f$ is not correctly specified in the question.