Referring to the graphs in my first post: for the left one, f(x) is continuous at x=2 because
1) the limit of f(x) as x approaches two exists
2) that limit is equal to the value of f at that point.
In symbols, .
The graph on the right is discontinuous at x=2 because, although the limit exists, it does not equal the value of the function at that point.
Informally continuity is often described as "it's continuous if you can draw it without lifting your pen," and applying that to the graph on the right, it is not continuous at x=2 because the function "jumps" and you have to lift your pen there.