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Lipschitz continuity is defined as follows: A function is Lipschitz continuous if there exists a $K \in \mathbb R$ such that

$|f(x) - f(y)| \leq K|x-y| \forall x,y \in D$

Now I was wondering if it is possible to say that if one function's Lipschitz constant is bigger than another function's, the first function's slope must also be bigger?

Markus Amalthea Magnuson
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TestGuest
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1 Answers1

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The slope, or derivative, of a differentiable Lipschitz-continuous function is bounded, in absolute value, by the function's Lipschitz constant. Functions with a larger Lipschitz constant can thus have a bigger slope. There may, however, also be points in the domain at which a function with a smaller Lipschitz constant has a bigger slope.

Note that the Lipschitz constant is a global property of a function, whereas the slope is a local concept.

Eckhard
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  • So, if I understand correctly, the maximally possible scope for a function with a bigger Lipschitz Constant over its whole range is bigger then for a function with a smaller constant (but that does not necessarily implies a bigger scope at each point, i.e. locally)? – TestGuest Mar 28 '13 at 23:23
  • That is right . – Eckhard Mar 28 '13 at 23:37