What is a Discontinuous Graph?

S3-SA5-0301

Grade Level:

Class 9

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

Definition

What is it?

A discontinuous graph is a graph that has breaks, jumps, or holes in it. You cannot draw a discontinuous graph without lifting your pen from the paper. It shows that the relationship between two quantities changes suddenly or has missing values.

Simple Example

Quick Example

Imagine a graph showing the price of an auto-rickshaw ride. If the auto driver charges a fixed rate for the first 2 km, then a different fixed rate for the next 2 km, and then a third rate, the graph of price vs. distance would have jumps. These jumps mean it's discontinuous because the price doesn't smoothly increase.

Worked Example

Step-by-Step

Let's plot the number of full cricket overs bowled in a match based on the time elapsed. --- Step 1: Assume the match starts at 10 AM (0 minutes) and 0 overs are bowled. (0,0) --- Step 2: After 30 minutes, 5 overs are completed. (30,5) --- Step 3: There's a rain break for 15 minutes. During this time, no overs are bowled, so the number of overs stays at 5. --- Step 4: After the break, at 45 minutes, 5 overs are still completed. (45,5) --- Step 5: The match resumes. At 75 minutes, 10 overs are completed. (75,10) --- Step 6: If you plot this, you'll see a horizontal line segment during the rain break, followed by a jump up to the next point where overs are bowled. The graph has a 'flat' part and then a sudden 'jump' in the rate of overs, making it discontinuous. --- Answer: The graph of overs bowled over time would be discontinuous due to the rain break where overs stopped increasing.

Why It Matters

Discontinuous graphs are important for understanding real-world situations like stock market crashes, changes in electricity tariffs, or how computer programs make decisions. Data scientists use them to spot sudden changes in data, and engineers use them to design systems that handle abrupt shifts, like traffic light timings or sensor readings.

Common Mistakes

MISTAKE: Thinking any graph with a sharp corner is discontinuous. | CORRECTION: A sharp corner (like in a V-shape) is still continuous if you can draw it without lifting your pen. Discontinuity means an actual break, jump, or hole.

MISTAKE: Confusing a graph that goes off the page with a discontinuous graph. | CORRECTION: A graph that goes infinitely up or down (like a vertical asymptote) is still continuous in the parts you can draw. Discontinuity is about a gap or jump at a specific point.

MISTAKE: Believing all real-world situations can be represented by continuous graphs. | CORRECTION: Many real-world scenarios, especially those involving discrete choices or sudden events (like a switch being turned on/off), are best represented by discontinuous graphs.

Practice Questions

Try It Yourself

QUESTION: Is the graph of the number of siblings a person has (0, 1, 2, 3...) versus their age continuous or discontinuous? | ANSWER: Discontinuous, because you can only have whole numbers of siblings, not fractions, and the number doesn't change smoothly with age.

QUESTION: A mobile data plan charges ₹100 for the first 1 GB and ₹50 for every additional GB. If you plot the cost versus data used, will the graph be continuous or discontinuous? Explain. | ANSWER: Discontinuous. There will be sudden jumps in cost at 1 GB, 2 GB, 3 GB, etc., as the price changes abruptly for each new gigabyte.

QUESTION: Imagine a graph showing the number of students in a classroom over a school day. At 10 AM, 30 students are present. At 10:30 AM, 5 students leave for a sports practice. At 11 AM, 2 more students join from another class. Sketch this graph and state if it's continuous or discontinuous. | ANSWER: Discontinuous. The graph will show sudden drops at 10:30 AM and a sudden jump at 11 AM, indicating breaks where the number of students changes abruptly.

MCQ

Quick Quiz

Which of the following scenarios would most likely result in a discontinuous graph?

The height of a plant growing over several months.

The temperature of water heating up on a stove.

The number of likes on a social media post increasing over time, but suddenly stopping for an hour due to server issues.

The distance covered by a car travelling at a constant speed.

The Correct Answer Is:

Option C describes a scenario where the 'likes' stop increasing for a period, creating a flat segment followed by a jump or a break in the rate of increase, making the graph discontinuous. The other options describe smooth, continuous changes.

Real World Connection

In the Real World

Think about how your internet speed changes. If your Wi-Fi signal drops suddenly, your download speed (measured in MBps) might go from high to zero, and then back up when the signal returns. A graph of your download speed over time would show these sudden drops and jumps, making it discontinuous. This is similar to how network engineers monitor system performance.

Key Vocabulary

Key Terms

BREAK: A gap in the graph where it stops and then restarts | JUMP: An abrupt change in value at a specific point, creating a vertical leap | HOLE: A single point missing from an otherwise continuous line | ASYMPTOTE: A line that a graph approaches but never touches, often indicating a discontinuity | STEP FUNCTION: A type of discontinuous graph that looks like a series of steps.

What's Next

What to Learn Next

Now that you understand discontinuous graphs, explore 'Continuous Graphs' to see the difference. Then, you can learn about 'Limits and Continuity' which are crucial concepts in higher mathematics, especially in calculus and understanding how functions behave at specific points.