How Can You Identify If a Function Is Discontinuous?
Understanding the discontinuous function is crucial for board exams and competitive tests, as questions often require you to identify where and why a function “breaks.” This maths concept also helps spot gaps and jumps in graphs, making it easier to solve real-world or piecewise problems confidently.
Formula Used in Discontinuous Function
A function f(x) is discontinuous at x = a if any of the following holds:
\( \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x) \), or
\( \lim_{x \to a} f(x) \neq f(a) \), or
f(a) is not defined.
Here’s a helpful table to understand discontinuous function more clearly:
Discontinuous Function Table
| Type of Discontinuity | Condition | Example |
|---|---|---|
| Removable | Limit exists but not equal to f(a), or f(a) undefined | \( f(x) = \frac{x^2 - 1}{x - 1} \) at \( x = 1 \) |
| Jump | Left and right limits exist but are not equal | Step function at integer points |
| Infinite | One or both limits infinite or not defined | \( f(x) = \frac{1}{x - 2} \) at \( x = 2 \) |
This table shows how each pattern of a discontinuous function arises and helps spot discontinuities during practice or board exams. For more, explore Types of Discontinuities and related graphs.
Worked Example – Solving a Problem
1. Given the function \( f(x) = \frac{3}{x - 2} \), check if it is discontinuous at \( x = 2 \).2. Check if \( f(2) \) is defined.
3. Since \( f(2) \) does not exist, the function is discontinuous at \( x = 2 \).
Final answer: Discontinuous at \( x = 2 \).
Many exam questions on discontinuous function require you to check both limits and function value at given points. Practice identifying jump, removable, and infinite discontinuities for thorough preparation. For more examples and concepts, visit Piecewise Functions or Step Function, which are typical sources of discontinuity.
Practice Problems
- Determine the points of discontinuity for \( f(x) = \frac{x + 1}{x - 3} \).
- Is the piecewise function \( f(x) = 2x \) for \( x < 1 \), \( f(x) = 3 \) for \( x = 1 \), \( f(x) = x + 1 \) for \( x > 1 \) continuous at \( x = 1 \)?
- List one example each of removable, jump, and infinite discontinuity with proper equations.
- Examine the function \( f(x) = \tan x \) for all discontinuities in the range \( 0 < x < 2\pi \).
Common Mistakes to Avoid
- Confusing discontinuous function with “not differentiable” (all discontinuous points are NOT always where a function is not defined; sometimes the value just doesn’t match).
- Missing jump discontinuities in step functions or piecewise functions.
- Not checking both left and right limits before concluding continuity/discontinuity.
Real-World Applications
The concept of a discontinuous function is used to model sudden changes, like switching prices or thresholds in a production system. Step changes in signals, or “jump” values in data science and statistics, often use functions with breaks. Vedantu demonstrates these ideas through industry, economics, and science scenarios. See more at step function
We explored discontinuous function definitions, formulae, worked examples, and real-world uses. By practising problems and learning to spot discontinuities, you build strong problem-solving skills for exams and daily maths. Keep practising on Vedantu for more confidence!
Explore related topics: limits and continuity, limits, continuity and differentiability, and limits of trigonometric functions using sandwich theorem to solidify your foundation.
FAQs on What Is a Discontinuous Function?
1. What is a discontinuous function?
A discontinuous function in mathematics is a function that is not continuous at one or more points in its domain. This means the graph of the function has breaks, jumps, or holes at those points, so you cannot draw the graph without lifting your pen.
2. How can I know if a function is discontinuous?
A function is discontinuous at a point if one or more of the following occur at that point:
• The function is not defined at the point.
• The left-hand and right-hand limits are not equal.
• The limit exists but does not equal the function’s value at that point.
Check these conditions by analyzing the function or its graph near the suspected discontinuity.
3. What is the difference between continuous and discontinuous functions?
Continuous functions have no breaks, jumps, or holes and can be drawn without lifting your pen. Discontinuous functions have one or more points where they break, jump, or have unmatched values, causing interruptions in their graphs.
4. What is an example of a discontinuous production function?
A discontinuous production function can be represented as a piecewise function like:
f(x) = { 10, if x < 5; 30, if x ≥ 5 }
This means output jumps suddenly from 10 to 30 when input x reaches 5, creating a jump discontinuity.
5. What are the types of discontinuities in functions?
The main types of discontinuities are:
• Removable Discontinuity: A hole in the graph where the function could be redefined to be continuous.
• Jump Discontinuity: The function suddenly jumps to a different value.
• Infinite Discontinuity: The function heads towards infinity, creating a vertical asymptote.
6. What is a removable discontinuity?
Removable discontinuity occurs at a point where the function has a 'hole,' but by redefining the function at that point, the discontinuity can be removed. It usually happens when the limit exists, but the function value is different or undefined at that point.
7. What is a jump discontinuity?
Jump discontinuity occurs when the left-hand and right-hand limits at a point exist but are not equal. This causes the graph to 'jump' from one value to another without connecting them smoothly.
8. What is an infinite discontinuity?
Infinite discontinuity happens when the function’s values increase or decrease without bound (approach infinity or negative infinity) as it nears a certain point, often resulting in a vertical asymptote.
9. Can discontinuous functions be graphed on Desmos or graphing calculators?
Yes, discontinuous functions can be plotted on Desmos and most graphing calculators. They will show areas with jumps, breaks, or holes. Make sure to use proper piecewise definitions or function notation to display discontinuities accurately.
10. Are discontinuous functions Riemann integrable?
Discontinuous functions can be Riemann integrable if the set of discontinuities has measure zero (like finite points or removable discontinuities). However, functions with too many or certain types of discontinuities may not be integrable using the Riemann approach.
11. Can you provide examples of discontinuous functions?
Examples of discontinuous functions include:
• Step function: f(x) = [x] where [x] is the greatest integer less than or equal to x.
• Piecewise definitions: f(x) = 1 for x < 0, f(x) = 2 for x ≥ 0.
• 1/x: Discontinuous at x = 0 (infinite discontinuity).
12. Where do discontinuous functions appear in real life?
In real life, discontinuous functions appear in situations involving sudden changes, such as:
• Switching electrical circuits on and off
• Ticket pricing (child/adult fares)
• Grading systems (pass/fail cutoffs)
These scenarios have outputs that change abruptly rather than smoothly.
