Short Introduction
Continuity is one of the most important concepts of Calculus. It helps us understand whether a function behaves smoothly around a point without any sudden breaks or jumps. In CBSE Class 12 Mathematics, Chapter 5 “Continuity and Differentiability” introduces students to the formal definition of continuity and its applications.
Exercise 5.1 focuses on checking the continuity of polynomial, rational, modulus, piecewise and trigonometric functions at specific points.
This complete solution guide by www.mymockmate.com provides easy-to-understand step-by-step explanations that are highly useful for CBSE Board Exams and competitive examinations.
Quick Information Box
| Particular | Details |
|---|---|
| Chapter | 5 |
| Chapter Name | Continuity and Differentiability |
| Exercise | 5.1 |
| Board | CBSE |
| Class | 12 |
| Difficulty Level | Easy to Moderate |
| Important For | Board Exams, JEE, CUET |
Concepts Used (Topics Covered)
โ Definition of Continuity
โ Left Hand Limit (LHL)
โ Right Hand Limit (RHL)
โ Existence of Function Value
โ Continuous Functions
โ Polynomial Functions
โ Rational Functions
โ Modulus Functions
โ Piecewise Functions
โ Trigonometric Functions
โ Greatest Integer Function
โ Composite Functions
Important Formulas
Definition of Continuity
A function f(x) is continuous at x = a if
lim xโa f(x) = f(a)
That is,
LHL = RHL = f(a)
Polynomial Functions
Every polynomial function is continuous for all real numbers.
Rational Functions
A rational function is continuous wherever its denominator is not zero.
Modulus Function
f(x)=|x|
is continuous for every real number.
Sum, Difference and Product Rule
If f(x) and g(x) are continuous at x=a, then
- f(x)+g(x)
- f(x)-g(x)
- f(x)รg(x)
are also continuous.
Question 1
1. Prove that the function f(x) = 5x โ 3 is continuous at x = 0, at x = โ 3 and at x = 5.
Question 2
Examine the continuity of the function f(x) = 2×2 โ 1 at x = 3.
Question 3
Examine the following functions for continuity.
Question 4
Prove that the function f(x) = xn is continuous at x = n, where n is a positive integer.
Find all points of discontinuity of f, where f is defined by
Discuss the continuity of the function f, where f is defined by
- Show that the function defined by g(x) = x โ [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
20. Is the function defined by f(x) = x2 โ sin x + 5 continuous at x = ฯ?
- Discuss the continuity of the following functions:
(a) f(x) = sin x + cos x
(b) f(x) = sin x โ cos x
(c) f(x) = sin x . cos x
22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.
31. Show that the function defined by f(x) = cos (x2) is a continuous function.
32. Show that the function defined by f(x) = |cos x| is a continuous function.
33. Examine that sin |x| is a continuous function.
34. Find all the points of discontinuity of f defined by f(x) = |x| โ |x + 1|.
Common Mistakes
โ Forgetting to check whether the function is defined.
โ Assuming every rational function is continuous everywhere.
โ Not comparing LHL and RHL in piecewise functions.
โ Ignoring denominator equal to zero.
โ Missing modulus function properties.
Exam Tips
โ Always write:
Function Value
โ
LHL
โ
RHL
โ
Conclusion
โ Remember:
Polynomial โ Continuous Everywhere
Rational โ Continuous except denominator = 0
Modulus โ Continuous Everywhere
Greatest Integer โ Discontinuous at Integers
Practice MCQs
1.
Which function is continuous everywhere?
A. 1/x
B. |x|
C. 1/(xโ2)
D. [x]
Answer: B
2.
The function 1/(xโ3) is discontinuous at
A. 1
B. 2
C. 3
D. 4
Answer: C
3.
Every polynomial function is
A. Discontinuous
B. Continuous
C. Undefined
D. None
Answer: B
4.
The greatest integer function is discontinuous at
A. Rational numbers
B. Irrational numbers
C. Integers
D. Positive numbers
Answer: C
FAQ Section
Q1. What is continuity?
A function is continuous if
LHL = RHL = Function Value.
Q2. Are polynomial functions always continuous?
Yes. Every polynomial function is continuous for all real numbers.
Q3. Where are rational functions discontinuous?
At points where the denominator becomes zero.
Q4. Is modulus function continuous?
Yes, |x| is continuous everywhere.
Q5. Which questions are most important for CBSE Boards?
Piecewise functions, rational functions and greatest integer functions.
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