Understanding Nonremovable Discontinuity in Mathematics: A Comprehensive Explanation with Examples

nonremovable discontinuity

In mathematics, a nonremovable discontinuity refers to a type of discontinuity in a function where the function cannot be made continuous by simply assigning a value to the point of discontinuity

In mathematics, a nonremovable discontinuity refers to a type of discontinuity in a function where the function cannot be made continuous by simply assigning a value to the point of discontinuity.

To better understand this concept, let’s consider an example. Suppose we have a function f(x) = (x^2 – 1) / (x – 1). In this case, we have a point of discontinuity at x = 1 since plugging in x = 1 into the function results in division by zero.

If we try to determine the limit of the function as x approaches 1, we can evaluate the left-hand limit and right-hand limit separately. By plugging in values slightly lower and higher than 1, we can observe the behavior of the function:

Limit as x approaches 1 from the left: lim(x -> 1-) [ (x^2 – 1) / (x – 1) ] = lim(x -> 1-) (x + 1) = 2.

Limit as x approaches 1 from the right: lim(x -> 1+) [ (x^2 – 1) / (x – 1) ] = lim(x -> 1+) (x + 1) = 2.

Both the left-hand and right-hand limits are equal to 2. However, the value of the function at x = 1 is undefined (division by zero). Therefore, we have a nonremovable discontinuity at x = 1.

In this example, no matter what value we assign to f(1) to “fill the hole,” we will not be able to make the function continuous at x = 1. This discontinuity is said to be nonremovable because it cannot be removed or fixed by changing the value of the function at that point.

In summary, a nonremovable discontinuity occurs when a function has a hole or undefined point where it is discontinuous, and assigning a value to that point does not make the function continuous.

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