- #1

Azael

- 257

- 1

Let the function

[tex]f:[0,\infty) \rightarrow \mathbb{R}[/tex] be lipschitz continous with lipschits constant K. Show that over small intervalls [tex] [a,b] \subset [0,\infty) [/tex] the graph has to lie betwen two straight lines with the slopes k and -k.

This is how I have started:

Definition of lipschits continuity [tex]|f(x)-f(y)| \leq k|x-y|[/tex]

[tex]b>a[/tex]

[tex]|f(b)-f(a)| \leq k(b-a) \Leftrightarrow -k(b-a) \leq f(b)-f(a) \leq k(b-a)[/tex]

But after this I am a bit stumped. I dont know how to continue:grumpy:

[tex]f:[0,\infty) \rightarrow \mathbb{R}[/tex] be lipschitz continous with lipschits constant K. Show that over small intervalls [tex] [a,b] \subset [0,\infty) [/tex] the graph has to lie betwen two straight lines with the slopes k and -k.

This is how I have started:

Definition of lipschits continuity [tex]|f(x)-f(y)| \leq k|x-y|[/tex]

[tex]b>a[/tex]

[tex]|f(b)-f(a)| \leq k(b-a) \Leftrightarrow -k(b-a) \leq f(b)-f(a) \leq k(b-a)[/tex]

But after this I am a bit stumped. I dont know how to continue:grumpy:

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