What is a Removable Discontinuity?

S3-SA5-0179

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A removable discontinuity in a function is like a tiny 'hole' in its graph that can be filled by simply defining the function at that single point. It occurs when the function is undefined at a specific point, but the limit of the function exists at that point.

Simple Example

Quick Example

Imagine you're tracking the price of a popular snack, say a packet of 'Bhujia'. Usually, the price is ₹10. But one day, the shopkeeper forgets to put a price tag on it, so for that one packet, the price is 'undefined'. However, everyone knows it should be ₹10. If we just 'define' that one packet's price as ₹10, the problem is solved. This 'missing price' is like a removable discontinuity.

Worked Example

Step-by-Step

Let's look at the function f(x) = (x^2 - 4) / (x - 2).

Step 1: Identify where the function is undefined. The denominator (x - 2) becomes zero when x = 2. So, f(2) is undefined.
---Step 2: Try to simplify the function. We know that (x^2 - 4) is a difference of squares, which can be factored as (x - 2)(x + 2).
---Step 3: Substitute the factored form back into the function: f(x) = [(x - 2)(x + 2)] / (x - 2).
---Step 4: For all x ≠ 2, we can cancel out the (x - 2) term from the numerator and denominator. This simplifies the function to f(x) = x + 2 (for x ≠ 2).
---Step 5: Now, find the limit of the simplified function as x approaches 2. Limit as x -> 2 of (x + 2) is 2 + 2 = 4.
---Step 6: Since the limit exists (it's 4) but the original function was undefined at x = 2, there is a removable discontinuity at x = 2.
---Answer: The function f(x) = (x^2 - 4) / (x - 2) has a removable discontinuity at x = 2.

Why It Matters

Understanding discontinuities is crucial in fields like AI/ML and Data Science, where models need to handle 'missing' or 'faulty' data points smoothly. Engineers use this to design systems that don't suddenly break down. It helps computer scientists write robust code that doesn't crash due to unexpected inputs.

Common Mistakes

MISTAKE: Thinking all undefined points are removable discontinuities. | CORRECTION: A discontinuity is only removable if the limit of the function exists at that point after simplification.

MISTAKE: Forgetting to check the limit after simplifying the function. | CORRECTION: Always calculate the limit of the simplified function as x approaches the point of discontinuity. If the limit is a finite number, it's removable.

MISTAKE: Confusing a removable discontinuity with a vertical asymptote. | CORRECTION: A vertical asymptote occurs when the limit approaches infinity, not a finite number, after simplification.

Practice Questions

Try It Yourself

QUESTION: Does the function g(x) = (x^2 - 9) / (x - 3) have a removable discontinuity? If yes, at what x-value? | ANSWER: Yes, at x = 3.

QUESTION: For the function h(x) = (x^2 - 1) / (x - 1), find the value that would 'fill' the hole at the discontinuity. | ANSWER: The value is 2.

QUESTION: Consider the function f(x) = (x^3 - 8) / (x - 2). Is there a removable discontinuity? If yes, what is the limit as x approaches the point of discontinuity? (Hint: x^3 - a^3 = (x - a)(x^2 + ax + a^2)) | ANSWER: Yes, at x = 2. The limit is 12.

MCQ

Quick Quiz

Which of the following functions has a removable discontinuity?

f(x) = 1 / x

g(x) = (x + 5) / (x - 5)

h(x) = (x^2 - 25) / (x + 5)

k(x) = |x| / x

The Correct Answer Is:

For h(x), (x^2 - 25) can be factored as (x - 5)(x + 5). So, h(x) simplifies to (x - 5) for x ≠ -5. The limit as x approaches -5 is -10, which is a finite number, making it a removable discontinuity. The other options have vertical asymptotes or jump discontinuities.

Real World Connection

In the Real World

In cricket analytics, if a sensor briefly fails to record a ball's speed, it creates a 'data hole'. If past and future speeds suggest a clear trend, we can 'fill' that missing data point using statistical methods, similar to how a removable discontinuity can be 'filled'. This helps provide smooth, continuous data for performance analysis and decision-making.

Key Vocabulary

Key Terms

DISCONTINUITY: A point where a function's graph has a break or jump. | LIMIT: The value a function approaches as the input approaches some value. | UNDEFINED: A point where a function does not have a specific output value. | FACTORIZATION: Breaking down an expression into a product of simpler terms.

What's Next

What to Learn Next

Great job understanding removable discontinuities! Next, you should explore 'Non-Removable Discontinuities' like jump and infinite discontinuities. This will give you a complete picture of all the ways a function can be 'broken' and how to identify each type.