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Suppose a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$ is concave and sufficiently smooth. We have:

$\|f(\mathbf{x})-f(\mathbf{x}_0)\|\le M\|\mathbf{x}-\mathbf{x}_0\|$ for some positive constant $M$.

If the $f$ is univariate, we know that $M$ is the absolute value of the slope of the tangent line at $\mathbf{x}_0$. But what is it for multivariable case, is there a special name in math given to it? Thanks a lot!

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consider $f:\mathbb R\to \mathbb R$ given by $f(x)=x$. It then holds that $|f(x)-f(x_0)|\le 17\cdot |x-x_0|$. It what sense is $17$ the slope of anything? –  Ittay Weiss Mar 6 '13 at 10:32
    
why not M=1? you know what I mean. And, if we take M=17, the inequality does not hold for all x. –  Zhou Heng Mar 6 '13 at 11:31
    
I don't follow. $M$ may not exist at all (consider $f(x)=-x^2$ for instance). Do you assume that $M$ exists? –  user1551 Mar 6 '13 at 12:18
    
why $M$ does not exist for $-x^2$? It is a convex function and the slope of the tangent line on the parabola satisfies the inequality for the fixed $x_0$ and all other $x$. –  Zhou Heng Mar 6 '13 at 12:24
    
I see the mistake in the post. I should add "absolute value of" –  Zhou Heng Mar 6 '13 at 12:26
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1 Answer 1

up vote 1 down vote accepted

This is the definition of Lipschitz continuity. $M$ is called the Lipschitz constant of $f$.

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Great! So, is sufficiently smooth concave function a special case of Lipschitz continuous function? –  Zhou Heng Mar 6 '13 at 12:35
    
The link I gave has examples and counter-examples. –  Learner Mar 6 '13 at 12:40
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