Understanding Point Discontinuity | Definition, Examples, and Solutions

Point Discontinuity

Point discontinuity, also known as a removable discontinuity or removable singularity, is a type of discontinuity that occurs when a function is undefined at a specific point but can be made continuous by assigning a value to that point

Point discontinuity, also known as a removable discontinuity or removable singularity, is a type of discontinuity that occurs when a function is undefined at a specific point but can be made continuous by assigning a value to that point.

In mathematical terms, a point discontinuity occurs at a specific x-value, where the function is undefined. Typically, this occurs when the value of the function at that point is not defined due to some kind of division by zero or an operation that produces an indeterminate form such as 0/0.

To better understand point discontinuity, let’s consider an example. Suppose we have the following function:

f(x) = (x^2 – 4) / (x – 2)

We can see that this function has a point discontinuity at x = 2 because the denominator becomes zero at that point. If we substitute x = 2 into the function, we get:

f(2) = (2^2 – 4) / (2 – 2) = 0/0

Here, the function is undefined at x = 2, resulting in a point discontinuity. However, we can make the function continuous by simplifying it or removing the factor causing the denominator to be zero:

f(x) = (x + 2)

By removing the factor of (x – 2) from the function, we can assign a value to f(2) = 4 and make the function continuous at x = 2.

In summary, a point discontinuity occurs when a function is undefined at a specific point but can be made continuous by assigning a value to that point. It is important to identify and handle point discontinuities appropriately when analyzing functions and their behavior.

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