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