Suppose that f : [a, b] → [a, b] is continuous. prove that f has a fixed point. that is, prove that there exists c ∈ [a, b] such that f (c) = c.

Step-by-step explanation:

define the function:

As both and x are continuous functions, will also be continuous.

Now, what can we say about ?

we know that , thus:

thus   is non-negative.

What about ? Again we have:

That means that is not positive.

Now, we can imagine two cases, either one of or is equal to zero, or none of them is. If either of them is equal to zero, we have found a fixed point! In fact, any point for which is a fixed point, because:

Now, if and , then we have that

and . And by Bolzano's theorem we can assert that there must exist a point c between a and b for which . And as we have shown before that point would be a fixed point. This completes the proof.