Thread: Proving a statement involving the intermediate value theorem

1. Proving a statement involving the intermediate value theorem

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.

2. What can be said about the values of $\displaystyle g(0)~\&~g(1)?$.

Recall that you are given $\displaystyle 0\le f(x) \le 1$.

3. I'm not sure what can be said. I don't know what f(x) is, I just know part of its domain and range. I'm also not sure how we are given $\displaystyle 0\le f(x) \le 0$

4. That was a typo. Plato meant to say $\displaystyle 0\le f(x)\le 1$. (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?

5. g(0) would be positive, g(1) would be negative.

6. Originally Posted by Glitch
g(0) would be positive, g(1) would be negative.
Correct. Now what?

7. Hmm, I'm not sure. I feel like I'm missing something obvious here...

8. $\displaystyle g(0)>0~\&~g(1)<0$.

Because $\displaystyle g$ is continuous apply the MVT.