Let f:[a,b] --> R be continous on [a,b] and differentiable on (a,b). Suppose that abs( f'(x) ) is less than 1 for all x in (a,b). Prove that f has at most one fixed point.
Let f:[a,b] --> R be continous on [a,b] and differentiable on (a,b). Suppose that abs( f'(x) ) is less than 1 for all x in (a,b). Prove that f has at most one fixed point.
Suppose $\displaystyle \alpha\,,\beta \in (a,b)$ are two fixed point of f(x). Apply now Rolle's theorem for f(x) in the interval $\displaystyle [\alpha,\beta]$ and get a contradiction.