Is it true that if $f(x,y)-f(x',y)$ is independent of $y$ for all y, then $f(x,y)-f(x',y)=g(x)-g(x')$ for some $g?$

Question

It is clear that it is true that $f(x,y)-f(x',y)=g(x,x'),$ but is the function $g$ necessarily a separable additive function? Thank you!