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Partial derivatives and convergence to zero

Functions with a zero derivative form an ideal of $C^infty(mathbb{R}^n)$$f: mathbb{R}^2to mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.Condition to satisfy sequence of bounded second derivativesWhen does convergence of function imply convergence of its derivative?What does bounded partial derivatives exactly mean?Is the partial derivative continuous w.r.t. other variables that locally Lipschitz continuous to the function?Convergence of second derivatives of uniformly convergent convex functionsWhen does $frac{partial f}{partial y} =0$ imply $f(x,y)=g(x)$?Bounded derivative of increasing functionClarity on total differentiability and partial differentiability

0

$begingroup$

Let $f(x)$ be a continuously differentiable function such that $f(x) to 0$ as $x to infty$. It is known that this does not necessarily imply that $df(x)/dx to 0$ as $x to infty$.

Consider now $g(y,x): D_y times D_x to mathbb{R}$, $D_y subseteq mathbb{R}$, $D_x subseteq mathbb{R}$. Assume that $g(y,x)$ is continuously differentiable, and $g(y,x) to 0$ as $xto infty$ $~~forall y in D_y$. What can be said about
$$lim_{xto infty} frac{partial g(y,x)}{partial y}.$$
Under what conditions will it converge to zero? Note that the partial derivative is taken w.r.t. $y$.

real-analysis functions convergence partial-derivative

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edited 2 days ago

Mateus de Freitas

asked 2 days ago

Mateus de FreitasMateus de Freitas

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add a comment | 

0

$begingroup$

Let $f(x)$ be a continuously differentiable function such that $f(x) to 0$ as $x to infty$. It is known that this does not necessarily imply that $df(x)/dx to 0$ as $x to infty$.

Consider now $g(y,x): D_y times D_x to mathbb{R}$, $D_y subseteq mathbb{R}$, $D_x subseteq mathbb{R}$. Assume that $g(y,x)$ is continuously differentiable, and $g(y,x) to 0$ as $xto infty$ $~~forall y in D_y$. What can be said about
$$lim_{xto infty} frac{partial g(y,x)}{partial y}.$$
Under what conditions will it converge to zero? Note that the partial derivative is taken w.r.t. $y$.

real-analysis functions convergence partial-derivative

share|cite|improve this question

edited 2 days ago

Mateus de Freitas

asked 2 days ago

Mateus de FreitasMateus de Freitas

32

$endgroup$

add a comment | 

$begingroup$

Let $f(x)$ be a continuously differentiable function such that $f(x) to 0$ as $x to infty$. It is known that this does not necessarily imply that $df(x)/dx to 0$ as $x to infty$.

Consider now $g(y,x): D_y times D_x to mathbb{R}$, $D_y subseteq mathbb{R}$, $D_x subseteq mathbb{R}$. Assume that $g(y,x)$ is continuously differentiable, and $g(y,x) to 0$ as $xto infty$ $~~forall y in D_y$. What can be said about
$$lim_{xto infty} frac{partial g(y,x)}{partial y}.$$
Under what conditions will it converge to zero? Note that the partial derivative is taken w.r.t. $y$.

real-analysis functions convergence partial-derivative

share|cite|improve this question

edited 2 days ago

Mateus de Freitas

asked 2 days ago

Mateus de FreitasMateus de Freitas

32

$endgroup$

share|cite|improve this question

edited 2 days ago

Mateus de Freitas

asked 2 days ago

Mateus de FreitasMateus de Freitas

32

share|cite|improve this question

edited 2 days ago

Mateus de Freitas

asked 2 days ago

Mateus de FreitasMateus de Freitas

32

asked 2 days ago

Mateus de FreitasMateus de Freitas

32

asked 2 days ago

Mateus de FreitasMateus de Freitas

32