Understanding Removable Discontinuity in Functions: Definition, Example, and Notation

Removable Discontinuity

also known as holes (common factor in numerator and denominator)

A removable discontinuity, also known as a removable singularity, is a type of discontinuity that occurs in a function where there is a hole in the graph at a certain point but can be filled by redefining the function at that point. Essentially, it means that the function could be made continuous if a single point were changed.

Here’s an example of a function with a removable discontinuity:

$$
f(x) = \begin{cases}
\frac{x^2 – 1}{x-1}, & x \neq 1 \\
2, & x = 1 \\
\end{cases}
$$

The numerator and denominator of the fraction both equal zero at $x=1,$ which creates a hole in the graph at that point. However, if we define $f(1) = 2,$ we can fill in the hole and make the function continuous at that point.

In terms of notation, we often use a limit to describe a removable discontinuity. If $f(x)$ has a removable discontinuity at $x=a,$ we can write:

$$
\lim_{x \to a} f(x) = L
$$

Where $L$ is the value that would fill in the hole if we were to redefine the function at $a.$ By definition, a removable discontinuity does not affect the limit of the function, so we can identify it because the limit exists but is different from the value of the function at that point.

Overall, a removable discontinuity is a type of discontinuity that occurs in a function where there is a single point of discontinuity that can be fixed by redefining the function at that point. It is commonly identified using a limit and does not affect the limit of the function.

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