Intuitively, a function is continuous if you can draw it without lifting your pen from the paper. In the diagram below, the function on the left is continuous throughout, but the function on the right is not. It is "discontinuous" at x = c.
Formally, a function is continuous at a point x = c if the (standard two-sided) limit exists there and is equal to the value of the function at c. In other words, if
f (x) = f (c) |
In order to show that
f (x) = f (c) |
you need to show that
f (x) |
exists, and
f (x) |
exists, and that they are both equal to f (c).