How to Remove Discontinuity from a Function

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.

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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 making sure theorems that need continuity, like the Intermediate Value Theorem, apply.
• Real-world Applications: In real-world scenarios, such as signal processing or data interpolation, removing discontinuities can be fundamental for creating smooth, predictable models.

Step-by-Step: Removing a Discontinuity

1. Find the trouble point. Look for an x-value where the function is undefined, such as a zero in the denominator, or where a piecewise function changes rules.
2. Find the limit there. For a rational function, factor the top and bottom, cancel the common factor, and substitute.
3. Check that it is removable. The limit must exist as a finite number, with both sides approaching the same value.
4. Redefine the function at that point so its value equals the limit.

Example 1: f(x) = (x² + x − 6)/(x − 2). Factor: (x + 3)(x − 2)/(x − 2) = x + 3 for x ≠ 2. The limit as x → 2 is 5. The redefined function is f(x) = (x² + x − 6)/(x − 2) for x ≠ 2, and f(2) = 5.

Example 2 (wrong value at the point): g(x) = x² for x ≠ 3, and g(3) = 4. The limit as x → 3 is 9, but g(3) = 4. Change the value to g(3) = 9, and g is continuous.

Finding k to Make a Function Continuous

Many problems hide the value you need behind a letter.

Example 3: f(x) = (x² − 25)/(x − 5) for x ≠ 5, and f(5) = k. Simplify to x + 5, so the limit at 5 is 10. Set k = 10.

Example 4: f(x) = ax + 1 for x < 2, and f(x) = x² + 3 for x ≥ 2. The right side gives 2² + 3 = 7 at x = 2. The left side approaches 2a + 1. Set 2a + 1 = 7, so a = 3.

Common mistake: Canceling terms instead of factors. In (x + 4)/(x + 2), you cannot cancel the x’s. Only a common factor such as (x − 2) can be canceled.

FAQ

What is a removable discontinuity in calculus?

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 matters for theorems that require continuity, such as the Intermediate Value Theorem, 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.

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How do you write the redefined function after removing a discontinuity?

Write it as a piecewise function. Keep the original rule for every x except the trouble point, and add a second line giving the limit value at that point. For (x² − 9)/(x − 3), write f(x) = (x² − 9)/(x − 3) when x ≠ 3, and f(x) = 6 when x = 3. You can also simply write f(x) = x + 3.

How do you find the value of k that makes a function continuous?

Find the limit of the function at the point where k appears, then set k equal to it. If the function is piecewise, make the left-hand and right-hand values match at the boundary and solve for k. For f(x) = kx for x < 1 and x + 4 for x ≥ 1, set k = 5.

What is the difference between a hole and a vertical asymptote?

Both can come from a zero in the denominator. If the factor cancels with a factor in the numerator, you get a hole, which is removable. If it does not cancel, the function grows without bound there, and you get a vertical asymptote, which is not removable. In (x − 2)/((x − 2)(x + 1)), there is a hole at x = 2 and an asymptote at x = −1.

Does removing a discontinuity change the graph?

Only at one point. The graph before has an open circle (a hole) at the trouble point, or a dot in the wrong place. After you redefine the function, that point is filled in on the curve. Everything else stays exactly the same.