have a continuous function f:[0,1] -> [0,1]

For some

For some

Using the Intermediate Value Theorem, show that there exists some x* such that f(x*)=x*

I have attempted the problem:

Make

. Then we are looking for the case when

. So if f(0)=0 or f(1)=1, then we are done. So we assume neither case is true. Then we have that

because f must take on a value in the interval [0,1], and the same for

. These two statements imply that

. So then by the intermediate value theorem, we can say that

for some

which means that

for some

As I am pretty rusty on my proof skills I was wondering if somebody could check this for me.

Thanks