A function f:A->R is called Lipschitz if there exists a bound M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x, y belong to A. Geometrically speaking a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f.
a- Show that if f:A->R is Lipschitz then it is uniformly continuous on A.
b- Is the converse statement true? Are all uniformly continuous functions necessarily Lipschitz?© BrainMass Inc. brainmass.com December 24, 2021, 5:07 pm ad1c9bdddf
f: A->R is Lipschitz, then there exists M>0, such that for any x,y in A, we have |f(x)-f(y)|<=M|x-y|. Now for any e>0, we can find ...
This is a proof regarding uniformly continuous and Lipschitz functions.