Section 2.3: 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).

One-Sided Continuity

Continuity Properties of Some Elementary Functions

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.