Are functions satisfying $|f(x)-f(y)|le L |nabla f(x)-nabla f(y)|^{1+s}$ constant?
$begingroup$
Let $f:mathbb R^nto mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such
$$
|f(x)-f(y)|le L |nabla f(x)-nabla f(y)|^{1+s} quad forall x,yinmathbb R^n.
$$
Does this imply that $f$ is constant?
Clearly if $nabla f$ is Lipschitz continuous (or $alpha$-Hoelder continuous with $alpha(1+s)>1$) then $nabla f=0$ follows immediately.
The question was inspired by this question Question about strong convexity, and my subsequent answer. There I show that also convexity of $f$ implis that $f$ is constant.
So the question is: are there non-constant $C^1$-functions satisfying the above inequality? Or is there a proof to show that a $C^1$ function satisfying the inequality is constant?
real-analysis
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb R^nto mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such
$$
|f(x)-f(y)|le L |nabla f(x)-nabla f(y)|^{1+s} quad forall x,yinmathbb R^n.
$$
Does this imply that $f$ is constant?
Clearly if $nabla f$ is Lipschitz continuous (or $alpha$-Hoelder continuous with $alpha(1+s)>1$) then $nabla f=0$ follows immediately.
The question was inspired by this question Question about strong convexity, and my subsequent answer. There I show that also convexity of $f$ implis that $f$ is constant.
So the question is: are there non-constant $C^1$-functions satisfying the above inequality? Or is there a proof to show that a $C^1$ function satisfying the inequality is constant?
real-analysis
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb R^nto mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such
$$
|f(x)-f(y)|le L |nabla f(x)-nabla f(y)|^{1+s} quad forall x,yinmathbb R^n.
$$
Does this imply that $f$ is constant?
Clearly if $nabla f$ is Lipschitz continuous (or $alpha$-Hoelder continuous with $alpha(1+s)>1$) then $nabla f=0$ follows immediately.
The question was inspired by this question Question about strong convexity, and my subsequent answer. There I show that also convexity of $f$ implis that $f$ is constant.
So the question is: are there non-constant $C^1$-functions satisfying the above inequality? Or is there a proof to show that a $C^1$ function satisfying the inequality is constant?
real-analysis
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
$endgroup$
Let $f:mathbb R^nto mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such
$$
|f(x)-f(y)|le L |nabla f(x)-nabla f(y)|^{1+s} quad forall x,yinmathbb R^n.
$$
Does this imply that $f$ is constant?
Clearly if $nabla f$ is Lipschitz continuous (or $alpha$-Hoelder continuous with $alpha(1+s)>1$) then $nabla f=0$ follows immediately.
The question was inspired by this question Question about strong convexity, and my subsequent answer. There I show that also convexity of $f$ implis that $f$ is constant.
So the question is: are there non-constant $C^1$-functions satisfying the above inequality? Or is there a proof to show that a $C^1$ function satisfying the inequality is constant?
real-analysis
real-analysis
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
asked Dec 11 '18 at 8:56
dawdaw
24.3k1645
24.3k1645
add a comment |
add a comment |
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