Find a function g(x) such that all the conditions of the fixed point theorem are satisfied on [0,1]
Lots of choices, such as
g(x) = 2|x - 1/2|
g(1/2) = 1/3
g(x) = x^2+sin(2πx)/2
g(0.419) = 0.419
g(x) = 2|x - 1/2|
g(1/2) = 1/3
g(x) = x^2+sin(2πx)/2
g(0.419) = 0.419