If a function f is continuous at x = a then we must have the following three conditions.

1. f(a) is defined; in other words, a is in the domain of f.
2. The limit
   

   must exist.

3. The two numbers in 1. and 2., f(a) and L, must be equal

Theorem A. A polynomial is continuous at each real number. A rational function is continuous at each point of its domain.

Theorem B. Suppose that f and g are functions which are continuous at the point x = a and suppose that k is a constant. Then

1. The product k f is continuous at x = a.
2. The sum f + g is continuous at x = a.
3. The difference f - g is continuous at x = a.
4. The product f g is continuous at x = a.
5. The quotient f / g is continuous at x = a provided that g(a) is not zero.

Theorem C. Suppose that g is a function which is continuous at x = a and suppose that f is a function which is continuous at x = g(a) then the composition of f and g is continuous at x = a.

Examples:

Example. Consider the function

The details are left to the reader to see

and

So we have

Since f(2) = 5, then f(x) is not continuous at 2.

Exercise 1. Find A which makes the function

 

continuous at x=1.

We have

and

So f(x) is continuous at 1 iff