Which Of The Functions Graphed Below Has A Removable Discontinuity

Which of the Functions Graphed Below Has a Removable Discontinuity?

Understanding discontinuities in functions is crucial in calculus and analysis. A discontinuity occurs when a function's graph is interrupted or broken. There are several types of discontinuities, including removable, jump, and infinite discontinuities. This article will delve into identifying removable discontinuities, focusing on how to visually and analytically determine which of several graphed functions exhibits this specific type of discontinuity. We'll explore the characteristics of removable discontinuities, compare them to other discontinuity types, and provide a robust methodology for accurate identification.

Understanding Discontinuities

Before we dive into removable discontinuities specifically, let's establish a clear understanding of discontinuities in general. A discontinuity occurs at a point c in the domain of a function f(x) if the function is not continuous at that point. Continuity at a point c requires three conditions to be met:

1. f(c) is defined: The function has a value at c.
2. lim_x→c f(x) exists: The limit of the function as x approaches c exists.
3. lim_x→c f(x) = f(c): The limit of the function as x approaches c equals the function's value at c.

If any of these conditions are not met, the function has a discontinuity at c. There are three main types of discontinuities:

1. Removable Discontinuity

A removable discontinuity occurs when the limit of the function as x approaches c exists, but it does not equal the function's value at c (or f(c) is undefined). Graphically, this manifests as a "hole" in the graph. The discontinuity can be "removed" by redefining the function at c to equal the limit.

Key Characteristics:

• Limit exists: lim_x→c f(x) = L, where L is a finite number.
• f(c) ≠ L (or f(c) is undefined): The function's value at c is either different from the limit or undefined.

2. Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and the right-hand limit of the function as x approaches c exist, but they are not equal. Graphically, this looks like a "jump" or a gap in the graph.

Key Characteristics:

• Left-hand limit exists: lim_x→c^- f(x) = L₁
• Right-hand limit exists: lim_x→c⁺ f(x) = L₂
• L₁ ≠ L₂: The left-hand and right-hand limits are different.

3. Infinite Discontinuity

An infinite discontinuity occurs when the limit of the function as x approaches c is either positive or negative infinity. Graphically, this often appears as a vertical asymptote.

Key Characteristics:

• Limit is infinite: lim_x→c f(x) = ±∞

Identifying Removable Discontinuities in Graphed Functions

To identify a removable discontinuity from a graph, look for the following:

1. A "hole" in the graph: The most obvious visual indicator is a missing point where the function appears to be continuous otherwise.
2. A defined limit: Even though there's a hole, the function appears to approach a specific value as x approaches the point of the discontinuity.

Analyzing the graph alone might be insufficient for definitive conclusions, especially in complex scenarios. Therefore, examining the function's analytical expression is essential to confirm the presence of a removable discontinuity.

Analytical Approach to Identifying Removable Discontinuities

Let's consider several examples of functions. To determine if a function has a removable discontinuity at a point c, follow these steps:

1. Find potential discontinuities: Look for values of x where the function is undefined (e.g., division by zero) or where there might be a change in the function's definition.
2. Evaluate the limit: Calculate the limit of the function as x approaches c from both the left (lim_x→c^- f(x)) and the right (lim_x→c⁺ f(x)). If both limits exist and are equal, the limit exists.
3. Check the function value: Determine if f(c) is defined and if it's equal to the limit calculated in step 2. If the limit exists but is not equal to f(c) (or f(c) is undefined), then the discontinuity at c is removable.

Example 1:

Consider the piecewise function:

f(x) = { x² - 1, if x ≠ 1 { 2, if x = 1

Let's analyze the point x = 1:

• lim_x→1 f(x) = lim_x→1 (x² - 1) = 1² - 1 = 0
• f(1) = 2

Since lim_x→1 f(x) = 0 ≠ f(1) = 2, the function has a removable discontinuity at x = 1. The hole can be filled by redefining the function at x = 1 as f(1) = 0.

Example 2:

Consider the function: f(x) = (x² - 4) / (x - 2)

This function is undefined at x = 2 (division by zero). Let's check the limit:

lim_x→2 f(x) = lim_x→2 (x² - 4) / (x - 2) = lim_x→2 (x - 2)(x + 2) / (x - 2) = lim_x→2 (x + 2) = 4

The limit exists and equals 4. However, f(2) is undefined. Therefore, this function has a removable discontinuity at x = 2. It can be removed by redefining the function as f(x) = x + 2 for all x (except possibly at a specific point which does not influence the function's nature).

Example 3: A function with a Jump Discontinuity

Consider the function:

f(x) = {x if x < 1 {2 if x >= 1

In this example, the left hand limit as x approaches 1 is 1, and the right hand limit is 2. Since these limits are unequal, there's a jump discontinuity, not a removable one.

Example 4: A function with an Infinite Discontinuity

Consider the function f(x) = 1/x

This has an infinite discontinuity at x = 0, as the limit as x approaches 0 is undefined (approaches positive infinity from the right and negative infinity from the left).

Comparing Removable, Jump, and Infinite Discontinuities

Feature	Removable Discontinuity	Jump Discontinuity	Infinite Discontinuity
Limit at c	Exists (finite value)	Exists (different left and right limits)	Does not exist (±∞)
Function value at c	Undefined or different from the limit	May or may not be defined	Undefined
Graphical representation	Hole in the graph	Jump in the graph	Vertical asymptote
Removable?	Yes	No	No

Conclusion

Identifying removable discontinuities requires a combined approach of visual inspection of the graph and a thorough analytical investigation of the function's behavior near the point of potential discontinuity. Understanding the distinctions between removable, jump, and infinite discontinuities is crucial for a complete comprehension of function behavior and for various applications in calculus and beyond. The examples provided illustrate how to systematically approach these types of problems, providing a framework for analyzing and classifying discontinuities accurately. Remember to always consider both the graphical representation and the analytical expression of the function for a comprehensive understanding.