How to Remove Discontinuity from a Function

Let’s explore this concept in more detail.

Understanding Removable Discontinuities

• Definition: A removable discontinuity at a point \( x = a \) occurs if the limit of \( f(x) \) as \( x \) approaches \( a \) exists, but either \( f(a) \) is not defined or \( f(a) \) is not equal to this limit.
• Graphical Representation: On a graph, a removable discontinuity might appear as a ‘hole’ or a ‘gap’ in the function.

Examples of Removable Discontinuities

1. Undefined at the Point:
   Function \( f(x) = \frac{(x^2 – 1)}{(x – 1)} \) is undefined at \( x = 1 \) but the limit as \( x \) approaches \(1\) exists.
2. Defined but not Matching the Limit:
   Function \( g(x) = \begin{cases}
   x + 2, & \text{if } x \neq 3 \\
   5, & \text{if } x = 3
   \end{cases} \) is defined at \( x = 3 \) but does not match the limit as \( x \) approaches \(3\).

Process of Removing Discontinuities

1. Identify the Point of Discontinuity: Determine the point \( x = a \) where the discontinuity exists.
2. Evaluate the Limit: Find the limit of the function as \( x \) approaches the point of discontinuity. Let this limit be \( L \).
3. Redefine the Function: Modify the function’s definition so that \( f(a) = L \). This step effectively ‘fills the hole’ or ‘bridges the gap’ in the function at the discontinuity.

Redefining Functions with Removable Discontinuities

• For Undefined Points: If \( f(a) \) was originally undefined, redefine ( f(a) ) to be the limit value. For example, redefine \( f(x) = \frac{(x^2 – 1)}{(x – 1)} \) at \( x = 1 \) to be \( f(1) = 2 \).
• For Defined but Inconsistent Points: If \( f(a) \) was defined but different from the limit, adjust the definition to match the limit. In the case of \( g(x) \), redefine \( g(3) \) to be \( 5 \).

Theoretical and Practical Implications

• Continuous Extension: The modified function, now continuous at the point of discontinuity, is often referred to as a “continuous extension” of the original function.
• Mathematical Analysis: This process is significant in mathematical analysis for simplifying functions and ensuring certain properties like integrability.
• Real-world Applications: In real-world scenarios, such as signal processing or data interpolation, removing discontinuities can be crucial for creating smooth, predictable models.

FAQ

What is a removable discontinuity in calculus?

A removable discontinuity occurs at a point in a function where the limit exists but the function either is not defined or does not match the limit at that point. It appears as a ‘hole’ or ‘gap’ in the graph of the function.

How can you identify a removable discontinuity?

A removable discontinuity can be identified by finding a point where the limit of the function exists, but the function is either not defined at that point or its value does not match the limit.

Can all discontinuities be removed?

No, only removable discontinuities can be removed. Other types of discontinuities, like jump or infinite discontinuities, cannot be eliminated by redefining the function.

How do you remove a removable discontinuity?

To remove a removable discontinuity, redefine the function at the point of discontinuity so that its value equals the limit of the function at that point.

Why is it important to remove discontinuities in calculus?

Removing discontinuities is important to make a function continuous at certain points. This can be crucial for certain mathematical operations like integration, and for applications in physics, engineering, and other fields where continuous functions are needed.

What is an example of removing a removable discontinuity?

Consider \( f(x) = \frac{(x^2 – 4)}{(x – 2)} \) for \( x \neq 2 \). The limit as \( x \) approaches \(2\) is \(4\), but \( f(2) \) is undefined. Redefine \( f(2) = 4 \) to remove the discontinuity.

Does removing a discontinuity change the overall behavior of a function?

Removing a discontinuity does not generally change the overall behavior of a function; it simply ‘fills in a gap’ to make the function continuous at a specific point.

Is redefining a function the only way to remove discontinuities?

For practical purposes, redefining the function at the discontinuity point is the standard method for removing removable discontinuities. There are no other commonly used methods for this purpose.

Can removing discontinuities affect the domain of a function?

Yes, removing a discontinuity can slightly alter the domain of the function by including points where the function was previously undefined.

Are removable discontinuities common in real-world applications?

Removable discontinuities are relatively common in theoretical mathematics but less so in real-world applications. In practical scenarios, functions are often defined to be continuous, or discontinuities are dealt with using other methods.