Let be a continuous, real function on the interval . Assume

Prove that there exists a real number such that .

* Proof. * Let . Then is continuous on since it is the difference of functions which are continuous on .

Then, and . If , then and we are done. Similarly, if , then and we are done as well.

Assume then that , and . Then

Hence, applying Bolzano’s theorem to , there is some such that . This implies , or

First of all, thank you for the great solutions for this book, it helps a lot.

you can prove it by the intermediate-value theorem instead of Bolzano’s theorem.

since g is continuous on [0,1] ,by the intermediate-value theorem, it takes on every value on [0,1] (including 0).

hence, we can find a number c such that g(c)=0 .

then, f(c)-c=0 -> f(c)=c.