Problem 6 - Calculus Early Transcendentals

Question

Sketch the graph of a function that has a jump discontinuity at \(x=2\) and a removable discontinuity at \(x=4,\) but is continuous elsewhere.

Step-by-Step Solution

Answer

Sketch a piecewise function: add jump at x=2, hole at x=4.

Step 1: Understand the Types of Discontinuities

A jump discontinuity is where a function has a sudden "jump" at a certain point. This means the left-hand limit and right-hand limit at that point do not equal each other. Meanwhile, a removable discontinuity occurs where a function is not continuous at a point, but can be made continuous by properly defining the function's value at that point.

Step 2: Choose a Basic Continuous Function

Start with a basic continuous function that is easy to manipulate, such as a linear function. For simplicity, let's choose the function \( f(x) = x \) over the domain \( x eq 2 \) and \( x eq 4 \).

Step 3: Introduce the Jump Discontinuity at \(x=2\)

Modify the function at \( x=2 \). Let’s define \( f(x) \) for \( x=2 \) such that the left-hand limit and right-hand limit are different. For example, let’s choose \( f(x) = x + 1 \) for \( x < 2 \) and \( f(x) = x - 1 \) for \( x > 2 \). This causes a jump of 2 units vertically at \( x=2 \).

Step 4: Introduce the Removable Discontinuity at \(x=4\)

To create a removable discontinuity at \( x=4 \), we define the function value \( f(x) \) for \( x=4 \) so that it does not equal the limit as \( x \) approaches 4. For instance, define \( f(4) = 0 \), but ensure that for \( x < 4 \) and \( x > 4 \), \( f(x) = x \), which would make the limit as \( x \to 4 \) equal to 4.

Step 5: Sketch the Graph

Draw the graph based on these definitions. For \( x < 2 \), plot a linear function just above the line \( y = x \), then a distinct point at 3 (\( f(2) = 3 \)) for the jump. After \( x=2 \), continue with \( y = x - 1 \) until just before \( x=4 \). Then leave a small "hole" at point \( (4, 4) \) to show the removable discontinuity. Put the point \( (4, 0) \) on the graph to indicate the defined value.

Key Concepts

Jump Discontinuity

In calculus, a jump discontinuity occurs when there is an abrupt change in the value of a function at a specific point. This happens because the left-hand limit and the right-hand limit of the function at that point are not equal. Imagine walking on a path and suddenly encountering a vertical step up or down. That's similar to a jump discontinuity in a graph.

If we have a function that is defined differently on either side of a point, it can cause this type of discontinuity. For instance, using the original problem, consider the function:

• For \( x < 2 \), the function is defined as \( f(x) = x + 1 \).
• For \( x > 2 \), it is \( f(x) = x - 1 \).

At \( x = 2 \), the function jumps from 3 to 1, resulting in a vertical gap at that point. Each side of the point follows a different rule, leading to the discontinuity.

Removable Discontinuity

A removable discontinuity is like finding a small hole on a path that can be easily patched up. It's a point on the graph where the function isn't continuous, but there's an easy fix to make it continuous. This can occur when the limit of the function as it approaches a specific point exists, but it doesn’t align with the function’s value at that point.

To visualize this, consider the example of the function from the original exercise:

• As \( x \) approaches 4 from either direction, the function \( f(x) = x \). This means the limit as \( x \to 4 \) is 4.
• However, the value of the function specifically defined at \( x = 4 \) is \( f(4) = 0 \).

The function thus has a removable discontinuity at \( x = 4 \) because we can "remove" this discrepancy by redefining \( f(4) \) to be 4, aligning it with the limit.

Graph Sketching

Graph sketching involves plotting the behavior of a function on a coordinate plane. This helps to visualize concepts like discontinuities. To graph the function with discontinuities as described in the original exercise:

• Begin by sketching the line \( y = x + 1 \) for \( x < 2 \), keeping it slightly above the line \( y = x \). Place a distinct point at (2, 3) to mark the end of this segment.
• Next, draw the line \( y = x - 1 \) for \( x > 2 \), beginning just after \( x = 2 \).
• Continue this line until just before \( x = 4 \), where you make a small open circle (hole) at \((4, 4)\) to indicate the removable discontinuity.
• Finally, plot the point (4, 0) to show the actual defined value at \( x = 4 \).

This method of graph sketching illustrates both the jump and removable discontinuities effectively, allowing for a clear understanding of their characteristics.