Understanding Removable Discontinuity: Filling the Holes in Math Functions

removable discontinuity

A removable discontinuity, also known as a removable singularity or a hole, is a type of discontinuity that can be “filled” to make the function continuous at that point

A removable discontinuity, also known as a removable singularity or a hole, is a type of discontinuity that can be “filled” to make the function continuous at that point.

In mathematics, a function is considered continuous if it has a smooth and unbroken graph, with no abrupt jumps, breaks, or holes. However, there are certain cases where a function may have a missing point or a hole in its graph, even though it is continuous everywhere else. This missing point is referred to as a removable discontinuity.

To understand what a removable discontinuity is, let’s consider an example. Suppose we have the function f(x) = (x^2 – 1)/(x – 1).

If we evaluate this function for values of x other than 1, we get a valid value. For example, if we plug in x = 2, we get f(2) = 3. Similarly, if we plug in x = 3, we get f(3) = 5. So, the function is continuous for all values of x except x = 1.

However, if we try to evaluate f(1), we get an indeterminate form, as division by zero is undefined. The numerator and denominator become both zero when x = 1. This implies that the graph of the function has a hole or a missing point at x = 1. We can visualize this by graphing the function, where the graph will appear continuous, except for a hole at x = 1.

To “fill” this hole and make the function continuous, we can redefine the function at x = 1. In this case, we can simplify the function by factoring the numerator as (x+1)(x-1)/(x – 1). Now, the common factor (x – 1) cancels out, and we are left with f(x) = x + 1.

The new function f(x) = x + 1 is defined everywhere and does not have any missing points or holes. Therefore, we have removed the discontinuity by redefining the function at the point where it was undefined.

This process of filling in the hole is what makes this type of discontinuity “removable.” By making a small adjustment or redefinition at the point of discontinuity, we can transform the function into one that is continuous throughout its domain.

It is important to note that not all discontinuities are removable. There are other types of discontinuities, such as jump discontinuities or essential discontinuities, which cannot be removed by a simple redefinition of the function at a single point.

In summary, a removable discontinuity occurs when there is a missing point or hole in the graph of a function that can be filled by redefining the function at that point. By removing the discontinuity, we can make the function continuous throughout its domain.

More Answers: