What is a Removable Discontinuity? | Simple Explanation for Class 9

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What is a Removable Discontinuity on a Graph?

Grade Level:

Class 9

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

Definition

What is it?

A removable discontinuity on a graph is like a tiny 'hole' in the function's path. It's a single point where the function is undefined, but if you 'fill' that hole, the graph would become smooth and continuous. You can 'remove' this discontinuity by redefining the function at that one point.

Simple Example

Quick Example

Imagine you're drawing a line with a pencil, but suddenly your pencil lifts for just a tiny second at one point, leaving a small gap, and then continues drawing the line perfectly. That tiny gap is like a removable discontinuity. If you just put a dot there, your line would be complete and smooth.

Worked Example

Step-by-Step

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

1. **Identify where the denominator is zero:** The denominator is (x - 2). If x - 2 = 0, then x = 2. This means the function is undefined at x = 2, so there's a discontinuity here.
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2. **Factor the numerator:** The numerator is x^2 - 4, which is a difference of squares. So, x^2 - 4 = (x - 2)(x + 2).
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3. **Rewrite the function:** Now, f(x) = [(x - 2)(x + 2)] / (x - 2).
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4. **Cancel common factors:** Since x is not equal to 2 (because the original function is undefined there), we can cancel out the (x - 2) term from the numerator and denominator. This leaves f(x) = x + 2 (for x not equal to 2).
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5. **Check the limit:** As x approaches 2, the value of x + 2 approaches 2 + 2 = 4. This means the 'hole' in the graph is at the point (2, 4).
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6. **Conclusion:** Since we could simplify the function by canceling a common factor and find a limit, the discontinuity at x = 2 is a removable discontinuity (a 'hole' at (2, 4)).

Why It Matters

Understanding removable discontinuities is crucial in fields like Data Science and Engineering, where you often deal with data that might have missing points or 'outliers.' In AI/ML, knowing how to 'smooth out' such issues helps in building more accurate models. It helps engineers design systems that function reliably even with small data gaps.

Common Mistakes

MISTAKE: Thinking all discontinuities are the same. | CORRECTION: Remember there are different types! A removable discontinuity is special because you can 'fill' the hole. Others, like jumps or asymptotes, cannot be easily fixed.

MISTAKE: Forgetting to check the limit when simplifying. | CORRECTION: After canceling common factors, always find what value the function approaches at the point of discontinuity. This tells you the 'y-coordinate' of the hole.

MISTAKE: Confusing a removable discontinuity with a vertical asymptote. | CORRECTION: A removable discontinuity happens when a factor in the denominator *cancels out* with a factor in the numerator. A vertical asymptote occurs when a factor in the denominator *does not* cancel out, making the function shoot off to infinity.

Practice Questions

Try It Yourself

QUESTION: Is there a removable discontinuity in the function g(x) = (x^2 - 9) / (x - 3)? If yes, where is the hole? | ANSWER: Yes, at x = 3. The hole is at (3, 6).

QUESTION: For the function h(x) = (x^2 - x) / x, identify the type of discontinuity at x = 0. | ANSWER: Removable discontinuity at x = 0. The hole is at (0, -1).

QUESTION: Consider the function f(x) = (x^3 - 8) / (x - 2). Does it have a removable discontinuity? If so, what are the coordinates of the 'hole'? (Hint: Factor the numerator using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)). | ANSWER: Yes, it has a removable discontinuity at x = 2. The hole is at (2, 12).

MCQ

Quick Quiz

Which of the following functions has a removable discontinuity at x = 1?

f(x) = 1 / (x - 1)

f(x) = (x - 1) / (x^2 - 1)

f(x) = (x + 1) / (x - 1)

f(x) = |x - 1|

The Correct Answer Is:

Option B, f(x) = (x - 1) / (x^2 - 1), can be simplified to f(x) = (x - 1) / [(x - 1)(x + 1)] = 1 / (x + 1) for x not equal to 1. This means the (x - 1) factor cancels out, indicating a removable discontinuity. The other options either have non-removable discontinuities (vertical asymptotes) or no discontinuity at x = 1.

Real World Connection

In the Real World

Imagine a sensor in a smart city traffic system that briefly glitches and misses a single data point about car speed. If engineers understand removable discontinuities, they can use mathematical models to 'fill in' that missing data point based on the surrounding data, ensuring the traffic light system continues to work smoothly without stopping due to a tiny error. This is like predicting a cricket score even if one ball's data is missing.

Key Vocabulary

Key Terms

DISCONTINUITY: A point where a function is not continuous, meaning its graph has a break or gap. | LIMIT: The value a function approaches as the input approaches a certain point. | FACTORING: Breaking down an expression into simpler parts (factors) that multiply together. | HOLE: The specific point on a graph where a removable discontinuity occurs.

What's Next

What to Learn Next

Great job understanding removable discontinuities! Next, you should explore 'Non-Removable Discontinuities' like jump discontinuities and vertical asymptotes. This will help you understand all the different ways a graph can have breaks and how they behave differently.