# Lipschitz continuity

• Nov 24th 2009, 09:14 AM
Lipschitz continuity
Let X and Y are Metric spaces and X is compact

f:X--> Y is locally Lipschitz continuous for every xo in X there exists a ball B(xo)

how to prove f is globally Lipschitz continuous for all x,y belongs X
st d(f(x),f(y))<=Ld(x,y)

Can anyone help with this problem.

Thank you
• Nov 24th 2009, 01:06 PM
Focus
Suppose $x \in X$ and let U_x be the Lipschitz n.hood of x, with Lipschitz constant L_x. What can you say about $\bigcup_{x\in X}U_x$? How can you pick the Lipschitz (global) constant?
• Nov 24th 2009, 01:41 PM