What is a removable discontinuity?

A removable discontinuity, also known as a hole, is a type of discontinuity present in a function that can be removed by defining the function at that point.

This type of discontinuity occurs when a function is not defined at a particular point but can be made continuous by defining the value of the function at that point. The function may have a different value at that specific point, but once the value is defined, the discontinuity is removed, and the function becomes continuous at that point.

Removable discontinuities are often found in rational functions when a denominator becomes zero, resulting in a hole in the graph. By simplifying the function or factoring out the common factors, the hole can be filled, and the function can be made continuous.

In mathematical terms, a removable discontinuity is characterized by a limit that exists at the point where the discontinuity occurs, allowing the function to be extended or redefined to fill the gap.

Overall, removable discontinuities are common in mathematical functions and can be easily resolved by defining the function at the point of discontinuity.