Strange property of the functional equation $f(x+y)=ax+by$.

Question

Suppose $$\tag{*} f(x+y)=ax+by $$ holds for some function $f:\mathbb{R}\to\mathbb{R}$ and some $a,b\in\mathbb{R}$. Given the function $g:\mathbb{R}^2\to\mathbb{R}$ defined as $$ g(x,y)=f(x+y) $$ we see that it is symmetric with respect to $x,y$, since $x+y=y+x$. Then, it follows that $$ ax+by=f(x+y)=g(x,y)=g(y,x)=f(y+x)=ay+bx $$ and thus $a=b$. Hence, if $a\neq b$, (*) does not hold for any function $f$. Is this correct? In some sense, symmetry imposed by $g(x,y)=f(x+y)$ is extended to $ax+by$.

Answer 1

Hint

Set $y=0$ and see that $f(x)=ax$.