Application of Intermediate Value Theorem for five-point formula (numerical different

I have a specific, for-learning-sake-only question on how the author of this link:

http://www.math.ucla.edu/~yanovsky/T..._solutions.pdf

gets past the details of the Intermediate Value Theorem on the following paragraph. If someone could fill in the details for me, it would be greatly appreciated because I'm having a hard time understanding.

"Note that the IVT was used above..."

Shouldn't it be

"Suppose is continuous on with

. Since ] is between [TEX]f^{(5)}(\xi_1)[\TEX] and , the Intermediate Value Theorem implies that a number exists between and , and hence in , with "?