How many fixed points are there?

Question

Let $f:[0,10)\rightarrow [0,10]$ be a continuous mapping.Then

$(A)f$ need not have any fixed point.

$(B)f$ has atleast 10 fixed points.

$(C)f$ has atleast 9 fixed points.

$(D)f$ has atleast one fixed points.

WHAT I THINK- Since $[0,10)$ under usual metric is not complete.So,by Banach fixed point theorem $f$ need not have any fixed point i.e.,(A) is true.

Please give your advice/suggestions/opinions about my choice.

If there is any other method by which we can deal with this problem,please share...

Answer 1

Just take $f(x)=10$ for all $x\in [0,10)$. As we drop $10$, there are no fixed points. This counterexample immediately proves that statement $A$ is true.

EDIT: Suppose the question was whether a continuous map $f:[0,10]\rightarrow [0,10]$ has fixed points, how could we solve it?

If $f(0)=0$ or $f(10)=10$ there are fixed points, so assume this is not the case. Consider the function $g(x)=f(x)-x$. Clearly $f(0)>0$ and $f(10)<0$. By the intermediate value theorem there must exist an $c\in (0,10)$ such that $g(c)=0$. Equivalently, $f(c)=c$.

So in this case you still don't need the machinery of Banach's fixed point theorem to solve the question.