Question 50
(SWE 3) The function $\varphi(x,y,z)$, defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers such that
$$\varphi(x,y,z)=f(x+y,z)=g(x,y+z)$$
for all real $x,y$, and $z$. Show that there is a function $h$ of one real variable such that
$$\varphi(x,y,z)=h(x+y+z)$$
for all real $x,y$, and $z$.
$$\varphi(x,y,z)=f(x+y,z)=g(x,y+z)$$
for all real $x,y$, and $z$. Show that there is a function $h$ of one real variable such that
$$\varphi(x,y,z)=h(x+y+z)$$
for all real $x,y$, and $z$.