Can A Removable Discontinuity Be A Local Maximum Or Minimum

Can a Removable Discontinuity Be a Local Maximum or Minimum?

The question of whether a removable discontinuity can be a local maximum or minimum is a fascinating one that delves into the intricacies of calculus and function analysis. While the intuitive answer might lean towards "no," a more rigorous examination reveals a nuanced truth that depends heavily on how we define "local maximum" and "local minimum" and how we handle the undefined point itself. Let's explore this in detail.

Understanding Removable Discontinuities

Before tackling the core question, let's solidify our understanding of removable discontinuities. A removable discontinuity occurs at a point where a function is undefined, but the limit of the function as x approaches that point exists. This means there's a "hole" in the graph that could be "filled" by redefining the function at that specific point. The function's behavior around this point is well-behaved; it approaches a specific value, but that value isn't actually attained because the function isn't defined there.

Example: Consider the function f(x) = (x² - 1) / (x - 1). This function is undefined at x = 1 because it leads to division by zero. However, if we factor the numerator, we get f(x) = (x - 1)(x + 1) / (x - 1). For x ≠ 1, we can simplify this to f(x) = x + 1. The limit as x approaches 1 is 2. Therefore, the discontinuity at x = 1 is removable; we could redefine f(1) = 2 to make the function continuous.

Defining Local Extrema

The terms "local maximum" and "local minimum" refer to points where a function attains a maximum or minimum value within a small neighborhood around that point. More formally:

• Local Maximum: A function f(x) has a local maximum at x = c if there exists an interval (a, b) containing c such that f(c) ≥ f(x) for all x in (a, b).

• Local Minimum: A function f(x) has a local minimum at x = c if there exists an interval (a, b) containing c such that f(c) ≤ f(x) for all x in (a, b).

Crucially, these definitions don't explicitly require the function to be defined at the point c. This subtle point is vital to addressing our main question.

Can a Removable Discontinuity Be a Local Extremum? The Case for "Yes"

Surprisingly, the answer is yes, a removable discontinuity can be a local maximum or minimum, albeit under a specific, carefully considered interpretation.

Consider a slightly modified version of our previous example: let's define a new function g(x):

g(x) = (x² - 1) / (x - 1) if x ≠ 1 g(x) = 1 if x = 1

In this case, we have deliberately assigned a value to g(1) that is different from the limit as x approaches 1 (which is 2). Observe that the limit of g(x) as x approaches 1 is still 2. However, in the immediate vicinity of x = 1, g(x) approaches 2, but g(1) = 1. Therefore, g(1) is strictly less than the values of g(x) in a small neighborhood around x = 1. This makes g(1) a local minimum, even though x=1 is a removable discontinuity.

Similarly, we can construct a function where the value assigned to the point of discontinuity is greater than the surrounding values, thus creating a local maximum.

Illustrative Example:

Let's consider a piecewise function h(x):

h(x) = -|x| if x ≠ 0 h(x) = 1 if x = 0

Here, at x=0, we have a removable discontinuity. The limit of h(x) as x approaches 0 is 0, but h(0) = 1. Thus h(0) is a local maximum. The function has a "spike" at x=0.

The Importance of the Definition and Neighborhoods

The key to understanding this seemingly paradoxical result lies in carefully examining the definitions of local extrema and how we consider the "neighborhood" around the point of discontinuity. The definitions explicitly state that the comparison of values needs to be made only for points within the neighborhood, excluding the point of the discontinuity itself.

The function's value at the discontinuity is independent from the limit value. We use the limit only to understand the behaviour of the function around the point. The critical point is that the function's behavior around the discontinuity (as defined by the limit) determines whether it’s a local extremum, while the actual value of the function at that point determines the value of the extremum.

The Case for "No" – Considering Continuous Extensions

One might argue against the possibility of a removable discontinuity being a local extremum if one focuses on the continuous extension of the function. If we "fill" the hole by redefining the function to be continuous at the point of discontinuity, the local extremum might disappear. In the continuous extension of the function g(x) above, where g(1)=2, the extremum disappears. The crucial point to note here is that the continuous extension is a different function from the original function.

This perspective highlights that the question's answer is partly a matter of whether we are focusing on the specific function with its discontinuity or its continuous extension.

Implications for Optimization Problems

The fact that removable discontinuities can represent local extrema carries implications for optimization problems. Standard optimization algorithms might miss these local extrema because they often rely on the differentiability of the function. The techniques for locating these extrema will generally require separate handling, potentially involving piecewise analysis or a careful consideration of the function's behavior at the point of discontinuity using limits.

Advanced Considerations: One-Sided Limits and Derivatives

For a more rigorous analysis, we can also consider one-sided limits and derivatives. The existence of a local extremum often involves the examination of the first derivative. However, when dealing with a removable discontinuity, the derivative might not be defined at the point of the discontinuity. Yet, the behavior of the one-sided derivatives in the neighborhood can still indicate the presence of a local maximum or minimum.

Conclusion: A Nuanced Answer

Ultimately, the question of whether a removable discontinuity can be a local maximum or minimum depends on the specific definition used and how the function is defined at the point of discontinuity itself. While a continuous extension of the function might not exhibit the local extremum, the original function with the discontinuity can indeed exhibit a local maximum or minimum at that point. This emphasizes the importance of careful function analysis and understanding the nuances of calculus definitions when dealing with optimization and function behavior. The distinction between the function's behavior around a point and its value precisely at that point is fundamental to this exploration.