Thread: sum of two lipschitz funtions

Let f : R ->R be a function.

We say that f is Lipschitz continuous if there is some
L > 0 such that |f(x) − f(y)| < L|x − y| for all x, y in R.


letting f, g : R -> R be lipschitz continious funtions,
(a) i want to show that f+g is lipschitz continious too

and also if f,g are bounded lip continuos funtions
(b) then i want to show that f.g is lipschitz continous too

does anyone know how to show these?

Originally Posted by dopi

Let f : R ->R be a function.

We say that f is Lipschitz continuous if there is some
L > 0 such that |f(x) − f(y)| < L|x − y| for all x, y in R.


letting f, g : R -> R be lipschitz continious funtions,
(a) i want to show that f+g is lipschitz continious too

Proof:

Since and are Lipschitz continuous, we have that for and for

Now let , and note that , then we have:



QED

you do (b), it should be similar

Originally Posted by dopi

Let f : R ->R be a function.

and also if f,g are bounded lip continuos funtions
(b) then i want to show that f.g is lipschitz continous too

If is Lipschitz then similarly so . We want to show for some . Note that where . Q.E.D.