When we say a function f is continuous, we usually mean it's continuous at every real number. In other words, it's continuous on the interval (-∞, ∞).
Some examples of continuous functions that are continuous at every real number are: polynomials, ex, sin(x), and cos(x).
If we add, subtract, multiply, or compose continuous functions, we find new continuous functions. If we take a quotient of continuous functions 
, this quotient will be continuous on any intervals that do not include places where g is zero. The quotient won't be defined there.