CMI-2020-DataScience-B: 19

Question

Let $f$ be a continuous bijection from closed unit interval $[0,1]$ onto itself. (Recall the Intermediate Value Theorem: let $f$ be a real valued continuous function on an interval $[a,b]$. Let $c,d\in [a,b]$ be such that $f(c)<f(d)$ and let $\alpha \in (f(c),f(d))$ be an intermediate value. Then there exists $y\in [a,b]$ such that $f(y)=\alpha.)$

1. Show that $f(0)$ equals $0$ or $1$.
2. Show that $f(1)$ equals $0$ or $1$.
3. Show that $f$ admits a fixed point.
4. Give an example of such a function where in the fixed point is unique and an example of a function with more than one fixed point.

Answer 1