TGT PGT Mathematics: Continuity Objective Questions 2025

TGT PGT Mathematics: Continuity Objective Questions

Year: 2025

Score: 0/50

1. A function f(x) is continuous at x = a if:

a) lim(x→a) f(x) exists b) f(a) is defined c) lim(x→a) f(x) = f(a) d) f(x) is differentiable at x = a

Correct Answer: c) lim(x→a) f(x) = f(a)
Explanation: A function is continuous at x = a if f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).

2. The function f(x) = |x| is continuous at:

a) x = 0 b) x = 1 c) All real numbers d) x ≠ 0

Correct Answer: c) All real numbers
Explanation: f(x) = |x| is continuous everywhere as lim(x→c) |x| = |c| = f(c) for all c.

3. If f(x) = 1/x, where is f(x) continuous?

a) x = 0 b) x ≠ 0 c) All real numbers d) x > 0

Correct Answer: b) x ≠ 0
Explanation: f(x) = 1/x is undefined at x = 0, hence continuous for x ≠ 0.

4. The function f(x) = [x] (greatest integer function) is continuous at:

a) All integers b) All non-integers c) All real numbers d) No real number

Correct Answer: b) All non-integers
Explanation: [x] has jump discontinuities at integers, so it’s continuous at non-integers.

5. If f(x) = {x² if x < 1, 2x-1 if x ≥ 1}, is f(x) continuous at x = 1?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 1² = 1, right limit = 2(1)-1 = 1, f(1) = 1. All equal, so continuous.

6. Which function is continuous everywhere?

a) f(x) = 1/x b) f(x) = x² + 2x + 1 c) f(x) = [x] d) f(x) = tan x

Correct Answer: b) x² + 2x + 1
Explanation: Polynomials are continuous everywhere.

7. The function f(x) = sin x is continuous:

a) Only at x = nπ b) For all real numbers c) Except at x = nπ/2 d) Except at x = 0

Correct Answer: b) For all real numbers
Explanation: Trigonometric functions like sin x are continuous everywhere.

8. If f(x) is continuous on [a, b], then f(x):

a) Is differentiable on [a, b] b) Attains its maximum and minimum c) Is increasing d) Is constant

Correct Answer: b) Attains its maximum and minimum
Explanation: By the Extreme Value Theorem, a continuous function on a closed interval attains its max and min.

9. The function f(x) = {x if x ≤ 0, x² if x > 0} is continuous at x = 0?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 0, right limit = 0² = 0, f(0) = 0. All equal, so continuous.

10. Which type of discontinuity does f(x) = 1/(x-2) have at x = 2?

a) Removable b) Jump c) Infinite d) None

Correct Answer: c) Infinite
Explanation: f(x) → ±∞ as x → 2, indicating an infinite discontinuity.

11. If f(x) = {2x if x < 2, 4 if x = 2, x+2 if x > 2}, is f(x) continuous at x = 2?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: b) No
Explanation: Left limit = 4, right limit = 4, but f(2) = 4. However, limits equal f(2), so correct answer is a) Yes.

12. If f(x) is continuous on [0, 1] and f(0) = f(1), then f(x) = c for some c in (0, 1):

a) Always true b) Never true c) True if f is differentiable d) True for some f

Correct Answer: d) True for some f
Explanation: By the Intermediate Value Theorem, f takes all values between f(0) and f(1), including c.

13. The function f(x) = tan x is continuous:

a) For all real numbers b) Except at x = nπ c) Except at x = (2n+1)π/2 d) Only at x = 0

Correct Answer: c) Except at x = (2n+1)π/2
Explanation: tan x is undefined at (2n+1)π/2, causing infinite discontinuities.

14. If f(x) = {x²-1)/(x-1) if x ≠ 1, 2 if x = 1}, is f(x) continuous at x = 1?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: lim(x→1) (x²-1)/(x-1) = lim(x→1) (x+1) = 2 = f(1), so continuous.

15. Which of the following is a removable discontinuity?

a) f(x) = 1/(x-3) at x = 3 b) f(x) = [x] at x = 2 c) f(x) = (x²-4)/(x-2) at x = 2 d) f(x) = tan x at x = π/2

Correct Answer: c) (x²-4)/(x-2) at x = 2
Explanation: lim(x→2) (x²-4)/(x-2) = 4, so defining f(2) = 4 removes the discontinuity.

16. If f(x) and g(x) are continuous at x = a, then f(x)g(x) is:

a) Continuous at x = a b) Not continuous at x = a c) Differentiable at x = a d) Constant at x = a

Correct Answer: a) Continuous at x = a
Explanation: The product of continuous functions is continuous.

17. The function f(x) = e^x is continuous:

a) Only at x = 0 b) For all real numbers c) Except at x = 0 d) Only for x > 0

Correct Answer: b) For all real numbers
Explanation: Exponential functions are continuous everywhere.

18. If f(x) = {sin x/x if x ≠ 0, 1 if x = 0}, is f(x) continuous at x = 0?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: lim(x→0) sin x/x = 1 = f(0), so continuous.

19. If f(x) is continuous on [a, b] and f(a)f(b) < 0, then:

a) f(x) is constant b) f(x) has a root in (a, b) c) f(x) is differentiable d) f(x) is increasing

Correct Answer: b) f(x) has a root in (a, b)
Explanation: By the Intermediate Value Theorem, f(x) = 0 for some x in (a, b).

20. The function f(x) = {x² if x rational, -x² if x irrational} is continuous at:

a) All real numbers b) x = 0 c) No real number d) All rationals

Correct Answer: b) x = 0
Explanation: At x = 0, f(0) = 0, and limits from rational and irrational approaches are 0. Elsewhere, limits don’t exist.

21. If f(x) = cos x, where is f(x) continuous?

a) Only at x = nπ b) For all real numbers c) Except at x = nπ/2 d) Except at x = 0

Correct Answer: b) For all real numbers
Explanation: Cosine is continuous everywhere.

22. If f(x) = {x²-4)/(x-2) if x ≠ 2, 4 if x = 2}, what type of discontinuity is at x = 2?

a) Jump b) Infinite c) Removable d) None

Correct Answer: c) Removable
Explanation: lim(x→2) (x²-4)/(x-2) = 4 = f(2), so the discontinuity is removable.

23. If f(x) is continuous at x = 0, and g(x) = f(x) + 3, is g(x) continuous at x = 0?

a) Yes b) No c) Depends on f d) Only if f is differentiable

Correct Answer: a) Yes
Explanation: Adding a constant preserves continuity.

24. The function f(x) = ln x is continuous for:

a) x > 0 b) x ≥ 0 c) All real numbers d) x ≠ 0

Correct Answer: a) x > 0
Explanation: ln x is defined and continuous for x > 0.

25. If f(x) = {1 if x rational, 0 if x irrational}, where is f(x) continuous?

a) All real numbers b) No real number c) Only at x = 0 d) Only at rationals

Correct Answer: b) No real number
Explanation: The Dirichlet function is discontinuous everywhere due to differing limits at rationals and irrationals.

26. If f(x) is continuous on [1, 3] and f(1) = 2, f(3) = 6, then f(c) = 4 for some c in (1, 3):

a) Always true b) Never true c) True if f is differentiable d) True for some f

Correct Answer: a) Always true
Explanation: By the Intermediate Value Theorem, f takes all values between 2 and 6, including 4.

27. If f(x) = {x² if x < 0, x if x ≥ 0}, is f(x) continuous at x = 0?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 0² = 0, right limit = 0, f(0) = 0. All equal, so continuous.

28. The function f(x) = 1/(x²-1) has what type of discontinuity at x = 1?

a) Removable b) Jump c) Infinite d) None

Correct Answer: c) Infinite
Explanation: f(x) → ±∞ as x → 1, indicating an infinite discontinuity.

29. If f(x) is continuous everywhere and g(x) = 1/f(x), then g(x) is continuous:

a) Everywhere b) Where f(x) ≠ 0 c) Where f(x) = 0 d) Nowhere

Correct Answer: b) Where f(x) ≠ 0
Explanation: g(x) is undefined where f(x) = 0, so continuous where f(x) ≠ 0.

30. The function f(x) = {x² if x ≤ 1, 2-x if x > 1} is continuous at x = 1?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: b) No
Explanation: Left limit = 1² = 1, right limit = 2-1 = 1, f(1) = 1. All equal, so continuous. (Correction: Answer is a) Yes)

31. If f(x) = x³ is continuous everywhere, then f⁻¹(x) is continuous:

a) Nowhere b) For all real numbers c) Only for x > 0 d) Only for x < 0

Correct Answer: b) For all real numbers
Explanation: f⁻¹(x) = x^(1/3) is continuous everywhere.

32. The function f(x) = {x if x rational, 0 if x irrational} is continuous at:

a) All real numbers b) x = 0 c) No real number d) All rationals

Correct Answer: b) x = 0
Explanation: At x = 0, f(0) = 0, and limits from rational and irrational approaches are 0. Elsewhere, discontinuous.

33. If f(x) = {x² if x < 2, 4 if x = 2, x+2 if x > 2}, is f(x) continuous at x = 2?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: b) No
Explanation: Left limit = 2² = 4, right limit = 2+2 = 4, f(2) = 4. All equal, so continuous. (Correction: Answer is a) Yes)

34. The function f(x) = csc x is continuous:

a) For all real numbers b) Except at x = nπ c) Except at x = (2n+1)π/2 d) Only at x = 0

Correct Answer: b) Except at x = nπ
Explanation: csc x = 1/sin x is undefined at sin x = 0, i.e., x = nπ.

35. If f(x) is continuous on [a, b], then f(x) is:

a) Differentiable on [a, b] b) Bounded on [a, b] c) Increasing on [a, b] d) Constant on [a, b]

Correct Answer: b) Bounded on [a, b]
Explanation: Continuous functions on closed intervals are bounded by the Boundedness Theorem.

36. The function f(x) = {x²-9)/(x-3) if x ≠ 3, 6 if x = 3} is continuous at x = 3?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: lim(x→3) (x²-9)/(x-3) = lim(x→3) (x+3) = 6 = f(3), so continuous.

37. If f(x) = {x if x < 1, 1 if x = 1, x² if x > 1}, is f(x) continuous at x = 1?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 1, right limit = 1² = 1, f(1) = 1. All equal, so continuous.

38. The function f(x) = sec x is continuous:

a) For all real numbers b) Except at x = nπ c) Except at x = (2n+1)π/2 d) Only at x = 0

Correct Answer: c) Except at x = (2n+1)π/2
Explanation: sec x = 1/cos x is undefined at cos x = 0, i.e., x = (2n+1)π/2.

39. If f(x) = {x² if x rational, x if x irrational}, where is f(x) continuous?

a) All real numbers b) x = 0 c) No real number d) All rationals

Correct Answer: b) x = 0
Explanation: At x = 0, f(0) = 0, and limits from rational and irrational approaches are 0. Elsewhere, discontinuous.

40. If f(x) is continuous on [0, 2] and f(0) = 1, f(2) = 3, then f(x) = 2 for some x in (0, 2):

a) Always true b) Never true c) True if f is differentiable d) True for some f

Correct Answer: a) Always true
Explanation: By the Intermediate Value Theorem, f takes all values between 1 and 3, including 2.

41. The function f(x) = {x²-16)/(x-4) if x ≠ 4, 8 if x = 4} is continuous at x = 4?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: lim(x→4) (x²-16)/(x-4) = lim(x→4) (x+4) = 8 = f(4), so continuous.

42. If f(x) = {x if x < 0, 0 if x = 0, x² if x > 0}, is f(x) continuous at x = 0?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 0, right limit = 0² = 0, f(0) = 0. All equal, so continuous.

43. The function f(x) = cot x is continuous:

a) For all real numbers b) Except at x = nπ c) Except at x = (2n+1)π/2 d) Only at x = 0

Correct Answer: b) Except at x = nπ
Explanation: cot x = cos x/sin x is undefined at sin x = 0, i.e., x = nπ.

44. If f(x) = {x² if x ≤ 0, x if x > 0}, is f(x) continuous at x = 0?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 0² = 0, right limit = 0, f(0) = 0. All equal, so continuous.

45. If f(x) is continuous on [a, b], then f(x) is:

a) Differentiable on [a, b] b) Uniformly continuous on [a, b] c) Increasing on [a, b] d) Constant on [a, b]

Correct Answer: b) Uniformly continuous on [a, b]
Explanation: Continuous functions on closed intervals are uniformly continuous.

46. The function f(x) = {x²-25)/(x-5) if x ≠ 5, 10 if x = 5} is continuous at x = 5?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: lim(x→5) (x²-25)/(x-5) = lim(x→5) (x+5) = 10 = f(5), so continuous.

47. If f(x) = {x if x < 2, 2 if x = 2, x² if x > 2}, is f(x) continuous at x = 2?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: b) No
Explanation: Left limit = 2, right limit = 2² = 4, f(2) = 2. Limits don’t match, so discontinuous.

48. The function f(x) = sinh x is continuous:

a) Only at x = 0 b) For all real numbers c) Except at x = 0 d) Only for x > 0

Correct Answer: b) For all real numbers
Explanation: Hyperbolic functions like sinh x are continuous everywhere.

49. If f(x) = {x² if x rational, 0 if x irrational}, where is f(x) continuous?

a) All real numbers b) x = 0 c) No real number d) All rationals

Correct Answer: b) x = 0
Explanation: At x = 0, f(0) = 0, and limits from rational and irrational approaches are 0. Elsewhere, discontinuous.

50. If f(x) = {x² if x ≤ 3, 9 if x = 3, x+6 if x > 3}, is f(x) continuous at x = 3?

a) Yes b) No c) Only from the left d) Only from the right

Correct Answer: a) Yes
Explanation: Left limit = 3² = 9, right limit = 3+6 = 9, f(3) = 9. All equal, so continuous.

TGT PGT Mathematics: Continuity Objective QuestionsYear: 2025

TGT PGT Mathematics: Continuity Objective Questions

Year: 2025

About the author

Chandra Shekhar

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TGT PGT Mathematics: Continuity Objective Questions

Year: 2025

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About the author

Chandra Shekhar

Leave a Comment X