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Statement 1:

A function $f$ satisfies a Lipschitz condition in the rectangular region $D$ if there is a positive real number $L$ such that $$|f(t, u) - f(t, v)| \leq L|u - v|$$ for all $(t,u) \in D$ and $(t, v) \in D$.

Statement 2:

There is a finite, positive, real number $L$ such that $$|\frac{d}{dy}f(t, y)| \leq L$$ for all $(t,y) \in D$.

Is this statement stronger than (i.e., more restrictive then), equivalent to or weaker than (i.e., less restrictive than) statement 1? Justify your answer.

My view:

I would say the statements are equivalent the max 'slope' for any two points in statement 1 won't exceed the max slope in statement 2.

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You can have a Lipschitz function not differentiable in one point. –  Tomás Nov 17 '12 at 14:01

1 Answer 1

up vote 1 down vote accepted

Statement 2 is stronger than Statement 1.

Define $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ by $$ f(t,u):=g(t)h(u),\text{ for all }(t,u)\in\mathbb{R}\times\mathbb{R} $$ where $g:\mathbb{R}\rightarrow\mathbb{R}$ is bounded and $h:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz. Then, $f$ satisfies the Statement 1 everywhere, whereas Statement 2 is satisfied only upto almost evereywhere (Rademacher Theorem).

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