Question

3. Suppose that f [0,1(0,1) is a non-decreasing function (NOT assumed to be continuous). Prove or disprove that there exists x E (0,1) such that f(x)-x

Answer 1

Intermediate value theorem let f be a function which is continuous on the closed interval [a, b] suppose that d is a real number between f(a) and f(b), then there exists c in [a, b] such that f(c) = d. Given that f: [0, 1] rightarrow [0, 1] be a continuous function we need to show that there exist x elementof [0, 1] such that f(x) = x Put g(x) = f(x) - x defined on [0, 1] Since f(x) is continuous and x is continuous [0, 1] Then g(x) = f(x) - x is continuous on [0, 1] Let f(0) = a, f(1) = b where a, b elementof [0, 1], so 0 lessthanorequalto b lessthanorequalto 1 Then g(0) = a > 0, g(1) = b - 1 < 0 Then g(0) g(1) < 0 By using intermediate value theorem, there exist c elementof [0, 1] such that g(c) = 0 Then f(c) - c = 0 Then f(c)