A function is continuous at a point if the limit of the function at the point is equal to the value of the function at the same point. In other words:
f is continuous at a <=> lim[x->a] f(x) = f(a)
To verify (i), look up the Intermediate Value Theorem.
To verify (ii), realize that a limit when x -> a exists if and only if limits when x ->a+ and x -> a- exist and are equal; what happens when they're different?
To verify (iii), realize that it is a particular case of: if f(x) is continuous at a point a, and g(v) is continuous at f(a), then g(f(x)) is continuous at a. To convince yourself, construct examples where f or g aren't continuous, and try to do the limit of g(f(x)).