What is Removable Discontinuity?

S7-SA1-0016

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

A removable discontinuity in a function is like a tiny 'hole' in its graph that can be 'filled' by redefining the function at just one point. It happens when the limit of the function exists at a certain point, but the function itself is either undefined or has a different value at that exact point.

Simple Example

Quick Example

Imagine you're tracking the price of a popular snack, say, a packet of 'bhujia'. For most days, the price is Rs 10. But on one particular day, the shopkeeper forgets to put a price tag, so for that one day, the price is 'undefined'. If you could just 'fill in' Rs 10 for that missing day, the price trend would be smooth again. That 'missing' price for one day 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: Try to find f(2). If we substitute x = 2, we get (2^2 - 4) / (2 - 2) = (4 - 4) / (0) = 0/0. This is an indeterminate form, meaning the function is undefined at x = 2.

---Step 2: Factorize the numerator. We know that a^2 - b^2 = (a - b)(a + b). So, x^2 - 4 = (x - 2)(x + 2).

---Step 3: Rewrite the function using the factored form: f(x) = [(x - 2)(x + 2)] / (x - 2).

---Step 4: For all values of x NOT equal to 2, we can cancel out the (x - 2) term from the numerator and denominator. So, f(x) = x + 2, for x not equal to 2.

---Step 5: Now, let's find the limit of f(x) as x approaches 2. Using the simplified form, lim (x->2) (x + 2) = 2 + 2 = 4.

---Step 6: Since the function is undefined at x = 2 (0/0 form), but the limit as x approaches 2 exists and is equal to 4, there is a removable discontinuity at x = 2.

---Step 7: We can 'remove' this discontinuity by redefining the function: g(x) = x + 2 for all x. This 'fills the hole' at x = 2 with the value 4.

Answer: The function f(x) = (x^2 - 4) / (x - 2) has a removable discontinuity at x = 2.

Why It Matters

Understanding removable discontinuities helps engineers design systems that run smoothly, like making sure a flight control system doesn't have 'gaps' in its data. In data science, knowing about these helps clean up messy data, ensuring AI models don't get confused by missing information. It's crucial for careers in software development, data analysis, and even designing safe vehicles.

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. If the limit doesn't exist (e.g., jumps or goes to infinity), it's a non-removable discontinuity.

MISTAKE: Forgetting to check if the function is actually defined at the point in question. | CORRECTION: For a removable discontinuity, the function must either be undefined at that point OR defined with a value different from the limit.

MISTAKE: Incorrectly simplifying the function before finding the limit. | CORRECTION: Always factorize and cancel terms carefully. Remember, you can only cancel (x-a) terms if x is NOT equal to a.

Practice Questions

Try It Yourself

QUESTION: Does the function f(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 g(x) = (x^2 - x - 6) / (x - 3), find the value that would 'fill the hole' at the removable discontinuity. | ANSWER: The value is 5.

QUESTION: Consider the function h(x) = (x^3 - 8) / (x - 2). Determine if it has a removable discontinuity and, if so, what is the 'hole's' y-coordinate? (Hint: Factor x^3 - a^3 as (x-a)(x^2+ax+a^2)) | ANSWER: Yes, it has a removable discontinuity at x = 2. The y-coordinate is 12.

MCQ

Quick Quiz

Which of the following functions has a removable discontinuity?

f(x) = 1/x at x = 0

g(x) = (x^2 - 1) / (x - 1) at x = 1

h(x) = |x|/x at x = 0

k(x) = tan(x) at x = pi/2

The Correct Answer Is:

Option B has a removable discontinuity because the function can be simplified to (x+1) for x not equal to 1, and the limit as x approaches 1 is 2. Options A, C, and D have non-removable discontinuities (infinite, jump, and infinite respectively).

Real World Connection

In the Real World

Imagine a sensor in an electric vehicle (EV) that monitors battery temperature. Sometimes, due to a tiny glitch, the sensor might miss a reading for a fraction of a second. If the temperature trend before and after this 'missing' point is smooth, engineers can use mathematical models to 'fill in' that missing data point. This ensures the EV's computer always has a complete picture, preventing sudden shutdowns or incorrect warnings, much like how ISRO scientists ensure continuous data flow from satellites.

Key Vocabulary

Key Terms

Discontinuity: A point where a function is not continuous, meaning its graph has a break or a jump. | Limit: The value that a function 'approaches' as the input approaches a certain value. | Indeterminate Form: Expressions like 0/0 or infinity/infinity, which don't immediately tell us the value. | Redefine: To change the definition of a function, usually at a single point, to make it continuous.

What's Next

What to Learn Next

Great job understanding removable discontinuities! Next, you should explore 'Non-Removable Discontinuities' (like jump or infinite discontinuities). This will help you fully grasp the different ways a function can be discontinuous and how to classify them, which is a key skill in calculus.