# Numerical theory and Lipschitz function

so I have this homework as I said and marks will be added on my total, so if anyone could help you will be a lifesaver, you dont have to answer the whole thing , just help me with the part you know,

here it is :
A function g (x) is called Lipschitz function on the interval [a,b] if there exists a constant L > 0, such that absolute(g(y) – g(x)) <= L *absolute(y-x).the constant L is called the Lipschitz constant.
1- Show that if g(x) is Lipschitz function on [a,b] with a Lipschitz constant L > 0, then g(x) is continuous function on [a,b].
2- Show that if g(x) is differentiable on [a,b], then g(x) is Lipschitz.
3- Show that if g(x) >= 0 is a Lipschitz function on [a,b], b > a >= 0 with a Lipschitz constant 0 < L =< 1, then g(x) maps the interval [a,b] into itself.
4- From the parts 1 and 2 , we deduct the existence of a fixed point P of g(x). show that P (the fixed point) is unique provided that g(x) is contraction function. A function g(x) is called a contraction function if g(x) is a Lipschitz function on [a,b] with a Lipschitz constant 0 < L < 1.
5- Assume that g(x) satisfies the condition in part 3 and 4. Show that the sequence of fixed point iterations defined by xn = g(xn-1) with any initial guess x0 converges to the unique fixed point.

HallsofIvy