TITLE: Can a function be continuous at the end points of its (closed interval) domain?
QUESTION [6 upvotes]: Assume $f$ has a domain of $[a, b]$. Is it possible that $f$ is continuous at $x = a$ and $x = b$?
If the definition of continuity is that the left and right limits are equal to the function at the given point, then this fails at $a$ since the left limit is undefined, and fails at $b$ since the right limit is undefined there. 
On the other hand there are other questions on this site that imply that it is possible for a function to be continuous on a closed interval, so perhaps this is simply an incorrect definition of continuity that I have seen in some high school text books.

REPLY [5 votes]: Your definition is correct only for interior points of the domain. For end points of the domain, the limit to be considered is the corresponding one-sided limit.
A couple of references.