The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function mapping the closed interval [0, 1] to itself must have a fixed

The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function ƒ mapping the closed interval [0, 1] to itself must have a fixed point; that is, a point c such that (c) = c.

(a) Give an example showing that if ƒ is continuous and 0 < ƒ(x) < 1 for 0 < x < 1, then there does not need to be a c in (0, 1) such that ƒ(c) = c.
(b) Give an example showing that if 0 ≤ ƒ(x) ≤ 1 for 0 ≤ x ≤ 1, but ƒ is not necessarily continuous, then there does not need to be a c in (0, 1) such that ƒ(c) = c.