Regarding a Feature of Multivariate Real Function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:19:47Z http://mathoverflow.net/feeds/question/48123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48123/regarding-a-feature-of-multivariate-real-function Regarding a Feature of Multivariate Real Function Wang Tao 2010-12-03T02:16:38Z 2010-12-03T11:51:44Z <p>Any real function can be expressed as a function of the sum of two monotonic real functions? </p> <p>More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such that:</p> <p>$p(x,y)=P(h(x,y)+g(x))$</p> <p>Where $P(x), h(x,y), g(x)$ are arbitrary satisfying $\frac {d(g(x))}{dx}>0$, $\frac {\partial h(x,y)}{\partial y}>0$ </p> <p>This is equivalent to mine another question “Solving Functional Equation”. By letting $h(x,y)=ln(w(x,y)), g(x)=-ln(u(x)), p(x,y)=\frac {u(f(x,y))}{u(x)}$, we have:</p> <p>$p(x,y)=\frac {u(f(x,y))}{u(x)}=F[ln(w(x,y))-ln(u(x))]=F(ln \frac {w(x,y)}{u(x)})=\Psi (\frac {w(x,y)}{u(x)})$</p> <p>Thank you very much!</p>