A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.
Sum, difference, product and quotient of continuous functions are continuous.
Every differentiable function is continuous, but the converse is not true.
Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0.