203 Real Mathematical Analysis Please provide complete and c

20.3. Real (Mathematical) Analysis. Please provide complete and correct solution done on computer or by hand with mathematical proof/explanation to all questions. Please, emphasis on complete and correct solution. The answers will be verified. This is all the information provided. Thank you very much.

3. Prove every continuous function f: [0, 1] lo, 1] has a fixed point (a point r such that f(r)

Solution

If f(0)=0 then we are done.

So, we assume that f(0)0, then 0<f(0).

Similarly, if f(1)=1, then we are done. So, we assume f(1)<1.

Consider another function h:[0,1]->[0,1] defined as h(x)=f(x)-x, which is also continuous as f(x) and x are both continuous.

h(0)=f(0)-0>0 (Since f(0)0) and h(1)=f(1)-1<0 (Since f(1)1)

Then, by Intermediate Value Theorem there exist c(0,1) such that h(c)=0, i.e. f(c)-c=0

So, f(c)=c.

Hence, every continuous function f:[0,1]->[0,1] has a fixed point.