Can A Removable Discontinuity Be A Local Maximum Or Minimum

Can a Removable Discontinuity Be a Local Maximum or Minimum?

A removable discontinuity, also known as a hole, occurs in a function when there's a single point where the function is undefined, but the limit of the function as x approaches that point exists. This article explores the intriguing question of whether a removable discontinuity can also be a local maximum or minimum. The short answer is: no, a removable discontinuity cannot be a local maximum or minimum. However, understanding why requires a deeper dive into the definitions of these concepts.

Let's break down the key concepts involved:

Understanding Removable Discontinuities

A removable discontinuity arises when there's a gap in the graph of a function at a specific point. This gap is "removable" because we could redefine the function at that single point to make it continuous. The function approaches a specific limit at the point of discontinuity, but the function itself is either undefined or defined differently at that point.

For example, consider the function:

f(x) = (x² - 1) / (x - 1)

This function has a removable discontinuity at x = 1. If we factor the numerator, we get:

f(x) = (x - 1)(x + 1) / (x - 1)

For x ≠ 1, we can simplify to f(x) = x + 1. The limit as x approaches 1 is 2, but f(1) is undefined. A simple redefinition, f(1) = 2, would remove the discontinuity.

Defining Local Maxima and Minima

A local maximum occurs at a point 'c' if f(c) is greater than or equal to f(x) for all x in some open interval containing 'c'. Similarly, a local minimum occurs at a point 'c' if f(c) is less than or equal to f(x) for all x in some open interval containing 'c'. Crucially, this definition requires the function to be defined at the point 'c'.

Why Removable Discontinuities Cannot Be Local Extrema

The core reason a removable discontinuity cannot be a local maximum or minimum is directly related to the definition of local extrema. For a point to be a local maximum or minimum, the function must be defined at that point. Since a removable discontinuity, by definition, has a gap (the function is undefined) at the point of discontinuity, it automatically fails the requirement for being a local extremum.

Even if the limit of the function at the discontinuity suggests a potential maximum or minimum value, the function's value at that point is either undefined or different, preventing it from satisfying the necessary conditions. The function must attain the value at the point in question to be classified as a local maximum or minimum.

Illustrative Example

Let's reconsider our example: f(x) = (x² - 1) / (x - 1). The limit as x approaches 1 is 2, which might seem like a local maximum or minimum. However, since f(1) is undefined, there's no local maximum or minimum at x = 1. The function simply approaches the value of 2 but never actually reaches it at that point.

In conclusion, while a removable discontinuity might appear to suggest a local maximum or minimum based on the limit, the undefined nature of the function at that point prevents it from being classified as such. The function must be defined at the point to be considered for local extrema. Therefore, the answer to the question is definitively no.