Intuitive definition
If the graph of the function can be drawn without "jumps," without having to remove the pencil from the paper, then it's continuous.
This works 99% of the time, but it can't be used as a proof that the function in question is continuous. However, it might be a handy trick when you're studying a graph.
In terms of limits
lim x -> c- f(x) = lim x -> c+ f(x) = f(c)
Using Weierstra** definition
For whatever ε > 0, given δ and in interval c - δ
f(c) - δ