2
$\begingroup$

A result I need is:

If $f(x,y) - f(x,z) = g(y,z)$ for all $(x,y,z)$, then $f(x,y) = a(x) + b(y)$ for some functions $(a,b)$.

This seems almost obvious, and I've constructed a proof, but that proof seems unnecessarily complicated and is remarkably tedious (and so isn't included here). I'd like pointers or ideas leading to something more simple and elegant. Surely this is a known result, or a special case of a well-known result?

$\endgroup$
  • $\begingroup$ What do you want to know by your statement ? $\endgroup$ – nmasanta May 31 at 17:38
4
$\begingroup$

Plugging $z=0$ into equation gives that $$ f(x,y)-f(x,0)=g(y,0), $$ so $$ f(x,y)=f(x,0)+g(y,0). $$ Now, just denote $a(x):=f(x,0)$ and $b(y):=g(y,0)$ and we get the desired result.

$\endgroup$
2
$\begingroup$

take $a(x)=f(x,z)$, $b(x)=g(x,z)$ for some $z$. then $f(x,y)=f(x,z)+g(y,z)=a(x)+b(y)$.

$\endgroup$
  • 1
    $\begingroup$ I think it's easier to understand if we choose some fixed value of $z$. $\endgroup$ – richrow May 31 at 17:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.