can a hole be a absolute maximum or minimum

Breiz Heurope

2 min read

·

10-03-2025

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The question of whether a hole can represent an absolute maximum or minimum in a function is a fascinating one that delves into the nuances of calculus and mathematical analysis. The short answer is: no, a hole (a removable discontinuity) cannot be an absolute maximum or minimum. However, understanding why requires a closer examination of the concepts involved.

Understanding Absolute Extrema

Before tackling holes, let's define absolute extrema. An absolute maximum is the largest value a function attains over its entire domain. Similarly, an absolute minimum is the smallest value. These points represent the highest and lowest peaks, respectively, on the function's graph.

The Nature of Holes (Removable Discontinuities)

A hole in a function's graph represents a removable discontinuity. This means there's a single point where the function is undefined, but the limit of the function as x approaches that point exists. In simpler terms, if you could "fill in" the hole, the function would be continuous at that point. The function approaches a specific y-value at the x-coordinate of the hole, but the function isn't actually defined at that x-value.

Why Holes Can't Be Extrema

The crucial distinction lies in the definition of absolute extrema. An absolute maximum or minimum must be a point where the function is actually defined. Since a hole represents a point where the function is undefined, it cannot be considered an absolute extremum. The function might approach a certain value at the hole, but it never actually reaches it at that precise point.

Illustrative Example:

Consider the function:

f(x) = (x² - 1) / (x - 1) for x ≠ 1

This function has a hole at x = 1. If we simplify the function, we get f(x) = x + 1 (for x ≠ 1). The limit as x approaches 1 is 2. However, f(1) is undefined. While the function approaches y = 2 near x = 1, it never actually attains the value of 2 at x = 1. Therefore, this point (1, 2), even though it seems like a potential minimum or maximum, is not an absolute extremum.

Local Extrema and Holes

While a hole cannot be an absolute extremum, it’s important to note that a function could have a local extremum near the location of the hole. A local maximum or minimum is the largest or smallest value in a small interval around a point. The function’s behavior near the hole could create a local maximum or minimum at a nearby point. However, the hole itself does not constitute this local extremum.

Conclusion: Holes and Extrema

In conclusion, a hole in a function's graph, being a point of discontinuity where the function is undefined, cannot be classified as an absolute maximum or minimum. The function might approach a value at the hole, suggesting a potential extremum, but the actual value isn't attained at that point due to the discontinuity. While local extrema might exist nearby, the hole itself remains outside the definition of extrema. Understanding this distinction is critical for a comprehensive grasp of calculus and function analysis.