The function
![\varphi(x,y,z)](/media/m/a/8/e/a8e716fa11abea1d516c9f1c0f5ee2e0.png)
defined for all triples
![(x,y,z)](/media/m/5/2/e/52e5ec762ff5263770c6e6c12cb9838e.png)
of real numbers, is such that there are two functions
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
defined for all pairs of real numbers, such that
for all real numbers
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
and
![z.](/media/m/9/1/6/91685beac3426769aff245bb3031d2cb.png)
Show that there is a function
![h](/media/m/e/4/3/e438ac862510e579cf5cbdbe5904d4ba.png)
of one real variable, such that
for all real numbers
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
and
%V0
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 numbers $x,y$ 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 numbers $x,y$ and $z.$