What can be said about the values of .
Recall that you are given .
The question:
Suppose that f is continuous on [0,1] and that Range(f) is a subset of [0,1]. By using g(x) = f(x) - x, prove that there is a real number c in [0,1] such that f(c) = c.
I'm not sure where to start. I know that the intermediate value theorem applies, since we're given a function g(x) which is continuous on the interval [0,1]. However, I don't know how to use this to prove that f(c) = c.
Any help would be great.
That was a typo. Plato meant to say . (That may be the first mistake he has ever made!)
First, if either f(0)=0 or f(1)= 1, we are done. So we can assume that f(0)> 0 and that f(1)< 1.
Let g(x)= f(x)- x as you say. Then g(0)= f(0). Is that positive or negative?
g(1)= f(1)- 1. Is that positive or negative?