Introduction to Calculus Precalculus Sixth Edition Chapter 11 Introduction to Calculus If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed: 1) MathType Plugin 2) Math Player (free versions available) 3) NVDA Reader (free versions available) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

11.3 Limits and Continuity

Learning Objectives Determine whether a function is continuous at a number. Determine for what numbers a function is discontinuous.

Definition of a Function Continuous at a Number A function f is continuous at a when three conditions are satisfied. ​​f is defined at a; that is, a is in the domain of f, so that f(a) is a real number. ​ ​

Example 1: Determining Whether a Function is Continuous at a Number (1 of 4) Determine whether the following function is continuous at 1: According to the definition, three conditions must be satisfied to have continuity at a. Condition 1 f is defined at a. Because f(1) is a real number, f(1) is defined.

Example 1: Determining Whether a Function is Continuous at a Number (2 of 4) Determine whether the following function is continuous at 1: According to the definition, three conditions must be satisfied to have continuity at a. Condition 1, that f is defined at a, has been satisfied Condition 2 Using properties of limits, we see that

Example 1: Determining Whether a Function is Continuous at a Number (3 of 4) Determine whether the following function is continuous at 1: According to the definition, three conditions must be satisfied to have continuity at a. Conditions 1 and 2 have been satisfied. Condition 3 Because the three conditions are satisfied, we conclude that f is continuous at 1.

Example 1: Determining Whether a Function is Continuous at a Number (4 of 4) Determine whether the following function is continuous at 2: According to the definition, three conditions must be satisfied to have continuity at a. Condition 1 f is defined at a. Factor the denominator of the function’s equation: f is not defined at 2. Therefore, f is not continuous at 2. We say “f is discontinuous at 2”.

Determining Where Functions are Discontinuous If f is a polynomial function, for any number a. A polynomial function is continuous at every number. Many functions are continuous at every number in their domain. Rational, exponential, logarithmic, sine, cosine, tangent, cotangent, secant, and cosecant functions are continuous at every number in their respective domains.

Example 2: Determining Where a Piecewise Function is Discontinuous (1 of 3) Determine for what numbers x, if any, the following function is discontinuous: First, we determine whether each of the three pieces of f is continuous. The first piece, a linear function, is continuous at every number x. The second piece, a quadratic function, is continuous at every number x. The third piece, a linear function, is continuous at every number x.

Example 2: Determining Where a Piecewise Function is Discontinuous (2 of 3) Determine for what numbers x, if any, the following function is discontinuous: We have determined that each of the three pieces of the function are continuous at x. We now investigate continuity at x = 0 and x = 2. We begin by investigating continuity at x = 0. Condition 1 f is defined at a. Because f(0) is a real number, f(0) is defined.

Example 2: Determining Where a Piecewise Function is Discontinuous (3 of 3) Determine for what numbers x, if any, the following function is discontinuous: We are investigating the continuity of f at x = 0. Condition 2 exists. The left- and right-hand limits are not equal. This means that the limit does not exist. The function is discontinuous at x = 0.