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.
Best Answer
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.