Continuous functions are, in essence, functions whose graphs can be drawn without lifting up your pen. This may sound simple, but this is in fact a very rich subject. Learn how continuity is defined using limits, and about a main property of all continuous functions -- the Intermediate value theorem.
The intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This is a basic but important property of all continuous functions.