Timeline for Regarding a Feature of Multivariate Real Function

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Dec 4, 2010 at 2:40	comment	added	Wang Tao		By setting $P(x)=e^x, g(x)=ln(x), h(x,y)=ln(y)$, we have: $p(x,y)=xy=e^{ln(xy)}=e^{ln(x)+ln(y)}=P(g(x)+h(x,y))$ Where: $\frac {\partial lnx}{\partial x}=\frac {1}{x}>0$, $\frac {\partial lny}{\partial y}=\frac {1}{y}>0$ for $x>0, y>0$.
Dec 3, 2010 at 12:00	comment	added	Willie Wong		I'm voting to close as the question is too localised. But a quick counterexample: Let $p(0,y)$ be some function such that $p(0,y) = 0$. Since $h$ is continuous and monotonic in $y$, there exists some $z_0 \in \mathbb{R}$ and a small $\delta$ such that $P(z) = 0$ whenever $\|z-z_0\| = \delta$, and such that there is some $(0,y_0)$ such that $h(0,y_0) + g(0) = z_0$. Now using continuity of $g, h$, the function $P(h(x,y) + g(x))$ must vanish on a neighborhood of $(0,y_0)$. So a function like $p(x,y) = xy$ cannot be written in the form you want.
Dec 3, 2010 at 11:57	comment	added	Joel David Hamkins		You seem to be missing a continuity hypothesis on $p$.
Dec 3, 2010 at 11:51	history	edited	Willie Wong		edited tags
Dec 3, 2010 at 2:23	history	edited	Wang Tao	CC BY-SA 2.5	added 5 characters in body; deleted 34 characters in body; added 1 characters in body
Dec 3, 2010 at 2:16	history	asked	Wang Tao	CC BY-SA 2.5