Problem 1 Fixed point recursion (10 + 10 = 20 points) A continuous function g : [a, b] + [a, b] must have at least one fixed point, that is, at least one solution x e [a, b] for the equation x = g(x). This is an immediate consequence of the intermediate value theorem (see Lecture 6, page 3). If in addition, we would like to prove that the fixed point is unique, that is, there is exactly one solution for x = g(x), then one useful result is the so-called contraction mapping theorem that says: if there exists 1 (0, 1) such that g(x) g(y)