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orl wrote: 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 $z.$ Trivial: $\varphi\left(0,x+y+z,0\right)=f\left(0+\left(x+y+z\right),0\right)=f\left(x+\left(y+z\right),0\right)=\varphi\left(x,y+z,0\right)$ $=g\left(x,\left(y+z\right)+0\right)=g\left(x,y+z\right)=\varphi\left(x,y,z\right)$. Thus, defining a function $h\left(u\right)=\varphi\left(0,u,0\right)$, we have $h\left(x+y+z\right)=\varphi\left(x,y,z\right)$, and the problem is solved. darij