Use the graph of f in the figure to do the following. a. Find the values of x in (-2,2) at which f is not continuous.

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.2.19a

Chapter 3, Problem 3.2.19a

Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (-2,2) at which f is not continuous.

Verified step by step guidance

Step 1: Understand the concept of continuity. A function f(x) is continuous at a point x = a if the following three conditions are met: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit of f(x) as x approaches a is equal to f(a).

Step 2: Examine the graph of f(x) over the interval (-2, 2). Look for any points where the function might not be continuous. These could be points where there is a jump, a hole, or a vertical asymptote in the graph.

Step 3: Identify any points of discontinuity. A point of discontinuity is where at least one of the conditions for continuity is not satisfied. Check for any breaks in the graph, such as gaps or jumps.

Step 4: Verify the type of discontinuity at each identified point. Determine if the discontinuity is removable (a hole in the graph), a jump (a sudden change in function value), or an infinite discontinuity (a vertical asymptote).

Step 5: List the x-values in the interval (-2, 2) where the function f(x) is not continuous, based on your observations from the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Discontinuities can occur due to jumps, holes, or vertical asymptotes in the graph.

Types of Discontinuities

There are several types of discontinuities: removable (or hole), jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by defining it appropriately. A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal, while an infinite discontinuity occurs when the function approaches infinity at a point.

Analyzing Graphs for Continuity

To determine where a function is not continuous using its graph, one must look for breaks, jumps, or asymptotes. Points where the graph does not connect smoothly indicate discontinuities. By examining the behavior of the graph as it approaches specific x-values, one can identify intervals where the function fails to be continuous.

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