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