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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.

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

1 Answer 1

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If $y=0$ then $f(x)=f(x+0)=f(0x)=f(0)$, so $f$ is a constant.

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  • $\begingroup$ Is your response answering the OP question? OP says: "is there a function that fulfills the condition:...."? $\endgroup$
    – NoChance
    Aug 29, 2019 at 20:15
  • $\begingroup$ @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. $\endgroup$ Aug 29, 2019 at 20:24
  • $\begingroup$ @DietrichBurde, I respect your point but I am not convinced. $\endgroup$
    – NoChance
    Aug 29, 2019 at 20:36
  • $\begingroup$ The problem here is that the domain of $f$ is not correctly specified in the question. $\endgroup$
    – IV_
    Aug 29, 2019 at 20:42

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