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When does the implicit function theorem guarantees that the set $g(x):={ymid f(x,y)=0}$ has only one element?
$begingroup$
Let $Xsubsetmathbb{R}^n$, $Ysubsetmathbb{R}^m$ be convex and $f:Xtimes Yrightarrow mathbb{R}^m$ be a continuously differentiable function such that for every $xin X$ there exits $yin Y$ such that $f(x, y)=0$.
I'm trying to find under which conditions I can guarantee that the set $g(x):={yin Ymid f(x, y)=0}$ has only one element for every $xin X$.
If $m=1$ I found that the condition is $frac{partial f}{partial y}>0$ for all $(x, y)in Xtimes Y$ (or $frac{partial f}{partial y}<0$), but I'm having problems when $m>1$.
implicit-function-theorem
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
$endgroup$
add a comment |
$begingroup$
Let $Xsubsetmathbb{R}^n$, $Ysubsetmathbb{R}^m$ be convex and $f:Xtimes Yrightarrow mathbb{R}^m$ be a continuously differentiable function such that for every $xin X$ there exits $yin Y$ such that $f(x, y)=0$.
I'm trying to find under which conditions I can guarantee that the set $g(x):={yin Ymid f(x, y)=0}$ has only one element for every $xin X$.
If $m=1$ I found that the condition is $frac{partial f}{partial y}>0$ for all $(x, y)in Xtimes Y$ (or $frac{partial f}{partial y}<0$), but I'm having problems when $m>1$.
implicit-function-theorem
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
$endgroup$
add a comment |
$begingroup$
Let $Xsubsetmathbb{R}^n$, $Ysubsetmathbb{R}^m$ be convex and $f:Xtimes Yrightarrow mathbb{R}^m$ be a continuously differentiable function such that for every $xin X$ there exits $yin Y$ such that $f(x, y)=0$.
I'm trying to find under which conditions I can guarantee that the set $g(x):={yin Ymid f(x, y)=0}$ has only one element for every $xin X$.
If $m=1$ I found that the condition is $frac{partial f}{partial y}>0$ for all $(x, y)in Xtimes Y$ (or $frac{partial f}{partial y}<0$), but I'm having problems when $m>1$.
implicit-function-theorem
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
$endgroup$
Let $Xsubsetmathbb{R}^n$, $Ysubsetmathbb{R}^m$ be convex and $f:Xtimes Yrightarrow mathbb{R}^m$ be a continuously differentiable function such that for every $xin X$ there exits $yin Y$ such that $f(x, y)=0$.
I'm trying to find under which conditions I can guarantee that the set $g(x):={yin Ymid f(x, y)=0}$ has only one element for every $xin X$.
If $m=1$ I found that the condition is $frac{partial f}{partial y}>0$ for all $(x, y)in Xtimes Y$ (or $frac{partial f}{partial y}<0$), but I'm having problems when $m>1$.
implicit-function-theorem
implicit-function-theorem
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
edited Jan 29 at 5:17
GuadalupeAnimation
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
asked Jan 29 at 5:10
GuadalupeAnimationGuadalupeAnimation
318110
318110
add a comment |
add a comment |
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