Definition types and solved examples of discontinuous function
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 Understanding Discontinuous Functions in Calculus
1. What is a discontinuous function?
A discontinuous function is a function that has at least one point where it is not continuous, meaning the limit does not equal the function’s value at that point. In other words, a function is discontinuous at x = a if:
- f(a) is not defined, or
- lim(x→a) f(x) does not exist, or
- lim(x→a) f(x) ≠ f(a).
2. What are the types of discontinuity in a function?
The main types of discontinuity are removable, jump, and infinite discontinuities. These include:
- Removable discontinuity: A hole in the graph where the limit exists but the function value is missing or different.
- Jump discontinuity: Left-hand and right-hand limits exist but are not equal.
- Infinite discontinuity: The function approaches ±∞ near a vertical asymptote.
3. How do you determine if a function is discontinuous at a point?
A function is discontinuous at x = a if it fails any condition of continuity at that point. To check:
- Step 1: Verify that f(a) is defined.
- Step 2: Compute lim(x→a) f(x).
- Step 3: Check if lim(x→a) f(x) = f(a).
4. What is an example of a discontinuous function?
An example of a discontinuous function is f(x) = (x² − 1)/(x − 1). Simplifying gives f(x) = x + 1 for x ≠ 1, but at x = 1 the function is undefined. Since lim(x→1) f(x) = 2 but f(1) is not defined, the function has a removable discontinuity at x = 1.
5. What is the difference between continuous and discontinuous functions?
A continuous function has no breaks in its graph, while a discontinuous function has at least one break, jump, or hole. Specifically:
- Continuous: lim(x→a) f(x) = f(a) for all points in its domain.
- Discontinuous: This equality fails at one or more points.
6. What is a removable discontinuity?
A removable discontinuity is a hole in the graph where the limit exists but the function value is missing or incorrect. It occurs when:
- lim(x→a) f(x) exists, and
- Either f(a) is undefined or not equal to the limit.
7. What is a jump discontinuity?
A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal. That means:
- lim(x→a⁻) f(x) exists,
- lim(x→a⁺) f(x) exists,
- But lim(x→a⁻) f(x) ≠ lim(x→a⁺) f(x).
8. What is an infinite discontinuity?
An infinite discontinuity occurs when the function approaches positive or negative infinity near a point. This happens when:
- lim(x→a) f(x) = ±∞,
- There is a vertical asymptote at x = a.
9. Can a function be discontinuous and still have a limit?
Yes, a function can be discontinuous at a point even if the limit exists. This occurs in a removable discontinuity where:
- lim(x→a) f(x) exists,
- But f(a) is either undefined or not equal to the limit.
10. Are rational functions always discontinuous?
Rational functions are continuous everywhere in their domain but discontinuous where the denominator equals zero. A rational function f(x) = p(x)/q(x) is discontinuous at values where q(x) = 0. These points create:
- Removable discontinuities if factors cancel, or
- Infinite discontinuities if they do not cancel.
