# Section 2.3: Continuity

## Continuity

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

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

### One-Sided Continuity

• 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 [a, b] if it is continuous on the open interval (a, b) and is continuous both on the right at a and on the left at b.

## Continuity Properties of Some Elementary Functions

• A constant function f(x) = k, where k is a constant, is continuous for all x.
• For n a positive integer, f(x) = xn 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.

## Sign Properties on an Interval (a, b)

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).

### Partition Numbers

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

### Constructing Sign Charts

Given a function f:

1. 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.
2. Plot the numbers found in step 1 on a real number line, dividing the number line into intervals.
3. 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.
4. Construct a sign chart, using the real number line in step 2. This will show the sign of f(x) on each open interval.