Continuity and Differentiability Practice Questions

Class 12 · Mathematics · 1849 free MCQs with instant results and detailed explanations.

1849

Total

581

Easy

910

Medium

358

Hard

Start Practicing Continuity and Differentiability

Take a timed quiz or customize your practice session

Quick Quiz (10 Qs) → Mock Test (25 Qs) ⚙ Customize

Topics in Continuity and Differentiability

Continuity 532

Implicit and Logarithmic Differentiation 477

Chain Rule 455

Differentiability 385

Sample Questions from Continuity and Differentiability

Here are 10 sample questions. Start a quiz to get randomized questions with scoring.

Easy

Which of the following functions is continuous at x = 2?

A. f(x) = x^2 - 4

B. f(x) = 1/(x - 2)

C. f(x) = sin(1/x)

D. f(x) = |x| for x < 0 and x for x ≥ 0

Show Answer & Explanation

Correct Answer: A

The function f(x) = x^2 - 4 is a polynomial function, which is continuous everywhere, including at x = 2. The others have discontinuities at or near x = 2.

Easy

For which value of k is the function f(x) = kx + 3 continuous at x = 1?

A. k = 0

B. k = -3

C. k = 1

D. Any value of k

Show Answer & Explanation

Correct Answer: D

The function f(x) = kx + 3 is linear and is continuous for all values of k. Any linear function does not have any points of discontinuity.

Easy

Which of the following statements about the continuity of the function g(x) = { x^2 for x < 1; 2 for x = 1; 3 for x > 1 } is true?

A. g(x) is continuous at x = 1

B. g(x) is discontinuous at x = 1

C. g(x) is continuous for all x

D. g(x) has a removable discontinuity at x = 1

Show Answer & Explanation

Correct Answer: B

To determine continuity at x = 1, we check the limit as x approaches 1. The limit is 1, but g(1) = 2. Hence, g(x) is discontinuous at x = 1.

Medium

Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 3x - 2, x >= 2 }

A. Continuous

B. Not continuous

C. Discontinuous at x=3

D. Continuous everywhere

Show Answer & Explanation

Correct Answer: A

The function is continuous at x=2 as the limits from both sides equal f(2).

Medium

If f(x) is continuous at x = a, which of the following statements must be true?

A. f(a) = limit as x approaches a of f(x)

B. f(a) is greater than limit

C. f(a) is less than limit

D. f(a) does not exist

Show Answer & Explanation

Correct Answer: A

For a function to be continuous at a point, the value at that point must equal the limit.

Medium

The function f(x) = x^3 - 3x + 1 is continuous everywhere. What is its value at x = 1?

A. 1

B. 0

C. 3

D. -1

Show Answer & Explanation

Correct Answer: A

Substituting x = 1 into the function gives f(1) = 1^3 - 3(1) + 1 = -1 + 1 = 1.

Medium

Which function has a removable discontinuity at x = 3? f(x) = (x^2 - 9)/(x - 3)

A. Has a removable discontinuity

B. Continuous

C. Discontinuous everywhere

D. Removable at x=0

Show Answer & Explanation

Correct Answer: A

The function has a removable discontinuity since it can be simplified to (x + 3) when x ≠ 3.

Hard

Let f(x) = x^3 - 3x^2 + 4. Determine the points of discontinuity, if any.

A. No points of discontinuity

B. x = 0

C. x = 1

D. x = 3

Show Answer & Explanation

Correct Answer: A

The function f(x) is a polynomial, which is continuous everywhere on the real line; hence, there are no points of discontinuity.

Hard

Consider the function f(x) defined as f(x) = sin(1/x) for x ≠ 0 and f(0) = 0. Is f(x) continuous at x = 0?

A. Yes

B. No

C. Only from the right

D. Only from the left

Show Answer & Explanation

Correct Answer: B

The limit as x approaches 0 does not exist because sin(1/x) oscillates indefinitely as x approaches 0, thus f is not continuous at x = 0.

Q10

Hard

If f(x) is continuous at x = c but not differentiable, which of the following must be true?

A. f(x) is constant

B. f(x) has a corner at x = c

C. f(x) has a vertical tangent

D. f(x) is differentiable everywhere

Show Answer & Explanation

Correct Answer: B

A corner in the graph of f(x) at x = c would mean it is continuous but not differentiable at that point.

Showing 10 of 1849 questions. Start a quiz to practice all questions with scoring and timer.

Practice All 1849 Questions →

More Mathematics Chapters

1 Relations and Functions

1660 Qs →

2 Inverse Trigonometric Functions

1949 Qs →

3 Matrices

2241 Qs →

4 Determinants

1944 Qs →

5 Application of Derivatives

1543 Qs →

6 Integrals

2245 Qs →

7 Application of Integrals

1878 Qs →

8 Differential Equations

2519 Qs →

Other Class 12 Subjects

🧬 Biology → 🧪 Chemistry → ⚡ Physics →

Continuity and Differentiability — Class 12 Mathematics Practice Questions Online

This page contains 1849 practice MCQs for the chapter Continuity and Differentiability in Class 12 Mathematics. The questions are organized by difficulty — 581 easy, 910 medium, 358 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit. This chapter covers 4 topics, giving you comprehensive coverage of the entire chapter.