A function *f* is **continuous at the point x = c if**:

**1.**limx → c*f(x)*exists.**2.***f(c)*exists.**3.**limx → c f(x) = f(c).

- If one or more of the three conditions fails, then the function is
**discontinuous**at x = c. - A function is
**continuous on the open interval (a, b)**if it is continuous at each point on the interval.

- A function is said to be
**continuous on the right**at*x = c*if limx → c^{+}f(x) = f(c). - A function is said to be
**continuous on the left**at*x = c*if limx → c^{-}f(x) = f(c). - A function is
**continuous on the closed interval [**if it is continuous on the open interval (*a, b*]*a, b*) and is continuous both on the right at*a*and on the left at*b*.

- A constant function f(x) = k, where
*k*is a constant, is continuous for all*x*. - For
*n*a positive integer, f(x) = x^{n}is continuous for all*x*. - A polynomial function is continuous for all
*x*. - A rational function is continuous for all
*x*except those values that make a denominator 0. - For
*n*an odd positive integer greater than 1,^{n}√ f(x) is continuous wherever*f(x)*is continuous. - For
*n*an even positive integer,^{n}√ f(x) is continuous wherever*f(x)*is continuous and nonnegative.

If *f* is continuous on (*a, b*) and f(x) ≠ 0 for all *x* in (*a, b*), then either f(x) > 0 for all *x* in (*a, b*) or f(x) < 0 for all *x* in (*a, b*).

A real number *x* is a **partition number** for a function *f* if *f* is discontinuous at *x* or f(x) = 0.

Given a function *f*:

- Find all partition numbers of
*f*. - Find all numbers
*x*such that*f*is discontinuous at*x*. Rational functions are discontinuous at values of*x*that make a denominator 0. - Find all numbers
*x*such that*f(x) = 0*. For a rational function, this occurs where the numerator is 0 and the denominator is not 0. - Plot the numbers found in step 1 on a real number line, dividing the number line into intervals.
- Select a test number in each open interval determined in step 2 and evaluate
*f(x)*at each test number to determine whether*f(x)*is positive (+) or negative (-) in each interval. - Construct a sign chart, using the real number line in step 2. This will show the sign of
*f(x)*on each open interval.