I've been trying to get my head round part of a proof and I don't see the following implication.
Suppose we have
then does it follow that
Since doesn't the above require that is continuous at , and if it does, why is this necessarily true?
The proof is from Trench's Introduction to Real Analysis, which can be freely downloaded here
William Trench - Trinity University Mathematics
I was wary of posting the entire proof verbatim in case of any infringement (not sure what the laws are concerning mathematical proofs but I thought it better to be safe).
The proof in question is on page 78 of the book.
EDIT: Actually I believe there may be a bracket missing from equation (10) in the proof, immediately after . Can anyone confirm this?