Continuity and Differentiability Class 12 Notes

• Left-hand limit and right-hand limit must both equal f(a).
• Sums, differences, products and quotients (with non-zero denominator) of continuous functions are continuous.
• A common composite rule:
• ${\lim }_{x\to a}f(g(x))=f\left({\lim }_{x\to a}g(x)\right)$
• when f is continuous.

• A function is differentiable on an open interval if it is differentiable at every point of it.
• If the left and right derivatives differ, the function is not differentiable (a sharp corner).
• Standard derivatives:
• $\frac{d}{dx}{x}^n=n{x}^{n-1}$
• ,
• $\frac{d}{dx}\sin x=\cos x$
• ,
• $\frac{d}{dx}{e}^x=e^x$
• .

• Chain rule:
• $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$
• .
• Product rule:
• $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
• .
• Quotient rule:
• ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$
• .
• Derivatives of inverse trig functions and the special forms
• $\frac{d}{dx}\ln x=\frac{1}{x}$
• ,
• $\frac{d}{dx}{a}^x=a^x\ln a$
• .

• Logarithmic differentiation: take ln of both sides first, then differentiate — ideal for
• $x^x$
• or long products and powers.
• Implicit differentiation: differentiate y(x) terms using the chain rule, then solve for
• $dy/dx$
• .
• Parametric form: if
• $x=f(t)$
• ,
• $y=g(t)$
• then
• $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$
• .
• The second derivative in parametric form:
• $\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)/\frac{dx}{dt}$
• (with care about the chain rule).

Example: Find $\frac{dy}{dx}$ if $y=\sin (x^2+1)$.

Solution: Using the chain rule with $u=x^2+1$: $\frac{dy}{dx}=\cos (x^2+1)\cdot 2x=2x\cos (x^2+1)$.

Exercises (NCERT Q1–Q121)

• 1) Prove that the function is continuous at , at and at .
• 2) Examine the continuity of the function at .
• 3) Examine the following functions for continuity: (a) ; (b) ; (c) ; (d) .
• 4) Prove that the function is continuous at , where n is a positive integer.
• 5) Is the function defined by continuous at , at , at ?
• 6) Find all points of discontinuity of where .
• 7) Find all points of discontinuity of where .
• 8) Find all points of discontinuity of where .
• 9) Find all points of discontinuity of where .
• 10) Find all points of discontinuity of where .
• 11) Find all points of discontinuity of where .
• 12) Find all points of discontinuity of where .
• 13) Find all points of discontinuity of where .
• 14) Discuss the continuity of the function where .
• 15) Discuss the continuity of the function where .
• 16) Discuss the continuity of the function where .
• 17) Find the relationship between and so that the function defined by is continuous at .
• 18) For what value of is the function defined by continuous at ? Also discuss the continuity of at .
• 19) Show that the function defined by is discontinuous at all integral points, where denotes the greatest integer function less than or equal to x.
• 20) Is the function defined by continuous at ?
• 21) Discuss the continuity of the following functions: (a) ; (b) ; (c) .
• 22) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
• 23) Find all points of discontinuity of defined by .
• 24) Determine if defined by is a continuous function.
• 25) Determine whether the function defined by is continuous.
• 26) Find the values of so that the function is continuous at .
• 27) Find the values of so that is continuous at .
• 28) Find the values of so that is continuous at .
• 29) Find the values of so that is continuous at .
• 30) Find the values of constants and such that the function is continuous.
• 31) Show that the function defined by is a continuous function.
• 32) Show that the function defined by is a continuous function.
• 33) Examine that is continuous.
• 34) Find all the points of discontinuity of defined by .
• 35) Differentiate with respect to x: .
• 36) Differentiate with respect to x: .
• 37) Differentiate with respect to x: .
• 38) Differentiate with respect to x: .
• 39) Differentiate with respect to x: .
• 40) Differentiate with respect to x: .
• 41) Differentiate with respect to x: .
• 42) Differentiate with respect to x: .
• 43) Prove that the function is not differentiable at .
• 44) Prove that the greatest integer function is not differentiable at and .
• 45) Find : .
• 46) Find : .
• 47) Find : .
• 48) Find : .
• 49) Find : .
• 50) Find : .
• 51) Find : .
• 52) Find : .
• 53) Find : .
• 54) Find : .
• 55) Find : .
• 56) Find : .
• 57) Find : .
• 58) Find : .
• 59) Find : .
• 60) Differentiate with respect to x: .
• 61) Differentiate with respect to x: .
• 62) Differentiate with respect to x: .
• 63) Differentiate with respect to x: .
• 64) Differentiate with respect to x: .
• 65) Differentiate with respect to x: .
• 66) Differentiate with respect to x: .
• 67) Differentiate with respect to x: .
• 68) Differentiate with respect to x: .
• 69) Differentiate with respect to x: .
• 70) Differentiate with respect to x: .
• 71) Differentiate with respect to x: .
• 72) Differentiate with respect to x: .
• 73) Differentiate with respect to x: .
• 74) Differentiate with respect to x: .
• 75) Differentiate with respect to x: .
• 76) Differentiate with respect to x: .
• 77) Differentiate with respect to x: .
• 78) Differentiate with respect to x: .
• 79) Differentiate with respect to x: .
• 80) Differentiate with respect to x: .
• 81) Find from .
• 82) Find from .
• 83) Find from .
• 84) Find from .
• 85) Differentiate and hence find .
• 86) Differentiate in three ways — (a) product rule, (b) by expanding, (c) by logarithmic differentiation. Verify that the answers agree.
• 87) Let u, v, w be functions of x. Prove that in two ways: by repeated product rule, and by logarithmic differentiation.
• 88) Find if .
• 89) Find if .
• 90) Find if .
• 91) Find if .
• 92) Find if .
• 93) Find if .
• 94) Find if .
• 95) Find if .
• 96) Find if .
• 97) Find if .
• 98) If , show that .
• 99) Find the second order derivative of .
• 100) Find the second order derivative of .
• 101) Find the second order derivative of .
• 102) Find the second order derivative of .
• 103) Find the second order derivative of .
• 104) Find the second order derivative of .
• 105) Find the second order derivative of .
• 106) Find the second order derivative of .
• 107) Find the second order derivative of .
• 108) Find the second order derivative of .
• 109) If , prove that .
• 110) If , find in terms of y alone.
• 111) If , show that .
• 112) If , show that .
• 113) If , show that .
• 114) If , show that .
• 115) If , show that .
• 116) Verify Rolle's theorem for .
• 117) Examine if Rolle's theorem is applicable to any of the following functions. What does this say about the converse of Rolle's theorem? (i) ; (ii) ; (iii) .
• 118) If is a differentiable function and if does not vanish anywhere, prove that .
• 119) Verify the Mean Value Theorem for in the interval [1,4].
• 120) Verify the Mean Value Theorem for in [1,3]. Find all for which the tangent is parallel to the chord.
• 121) Examine the applicability of the Mean Value Theorem for the three functions of question 2: (i) ; (ii) ; (iii) .

Answers at a glance

1. 1
   Q1 → Continuous at all three points.
2. 2
   Q2 → Continuous at x = 3, since f(3) = 17 = LHL = RHL.
3. 3
   Q3 → All four are continuous on their domains. (a) polynomial; (b) rational, continuous for x ≠ 5; (c) equals x − 5 for x ≠ −5, a removable discontinuity at x = −5; (d) |x − 5| is continuous everywhere.
4. 4
   Q4 → f(n) = nⁿ; as a polynomial, f is continuous at every real x, in particular at x = n.
5. 5
   Q5 → Continuous at x = 0 and x = 2; discontinuous at x = 1, since LHL = 1 but RHL = 5.
6. 6
   Q6 → Discontinuous at x = 2 (LHL = 7, RHL = 1).
7. 7
   Q7 → Discontinuous at x = −3 and x = 3.
8. 8
   Q8 → No point of discontinuity: |x| is continuous everywhere, including x = 0 where the limit is 0 = f(0).
9. 9
   Q9 → Discontinuous at x = 0, since LHL = −1 and RHL = 1.
10. 10
    Q10 → Continuous at x = 1 (LHL = 2 = RHL = f(1) = 2); no discontinuity.
11. 11
    Q11 → Continuous at x = 2: LHL = 8 − 3 = 5, RHL = 4 + 1 = 5, f(2) = 5. No discontinuity.
12. 12
    Q12 → Discontinuous at x = 1: LHL = 0 but RHL = 1.
13. 13
    Q13 → Discontinuous at x = 1: LHL = 6 but RHL = −4.
14. 14
    Q14 → Discontinuous at x = 1 and x = 3 (jumps of 1); continuous everywhere else.
15. 15
    Q15 → Continuous at x = 0 (both limits 0); discontinuous at x = 1, where LHL = 0 but RHL = 4.
16. 16
    Q16 → Continuous everywhere: at x = −1 both limits equal −2; at x = 1 both equal 2.
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → No value of λ makes f continuous at x = 0, because LHL = 0 while RHL = 1. At x = 1, f(x) = 4x + 1 is continuous.
19. 19
    Q19 → g is discontinuous at every integer and continuous at every non-integer.
20. 20
    Q20 → Yes. f(π) = π² + 5; since x² and sin x are continuous, f is continuous at x = π.
21. 21
    Q21 → All three are continuous on ℝ: sums, differences and products of the continuous functions sin x and cos x.
22. 22
    Q22 → cos x is continuous on ℝ. cosec x is continuous on ℝ − {nπ}; sec x on ℝ − {(2n+1)π/2}; cot x on ℝ − {nπ}, n ∈ ℤ.
23. 23
    Q23 → No point of discontinuity. At x = 0, LHL = lim (sin x)/x = 1, RHL = f(0) = 1, so f is continuous at 0.
24. 24
    Q24 → f is continuous everywhere. At x = 0, |x² sin(1/x)| ≤ x² → 0, so the limit is 0 = f(0).
25. 25
    Q25 → Discontinuous at x = 0: f(0) = 1, but LHL = RHL = sin 0 − cos 0 = −1.
26. 26
    Q26 → answer not in source — verify against the textbook
27. 27
    Q27 → answer not in source — verify against the textbook
28. 28
    Q28 → answer not in source — verify against the textbook
29. 29
    Q29 → answer not in source — verify against the textbook
30. 30
    Q30 → answer not in source — verify against the textbook
31. 31
    Q31 → cos is continuous and x² is continuous; a composition of continuous functions is continuous.
32. 32
    Q32 → cos x is continuous, and the modulus function is continuous; a composition of continuous functions is continuous.
33. 33
    Q33 → sin|x| = (sin ∘ |·|)(x). Both sin x and |x| are continuous, so the composite sin|x| is continuous on ℝ.
34. 34
    Q34 → Discontinuous at x = 0 and x = −1, where the two modulus terms change slope and the derivative jumps.
35. 35
    Q35 → answer not in source — verify against the textbook
36. 36
    Q36 → answer not in source — verify against the textbook
37. 37
    Q37 → answer not in source — verify against the textbook
38. 38
    Q38 → answer not in source — verify against the textbook
39. 39
    Q39 → answer not in source — verify against the textbook
40. 40
    Q40 → answer not in source — verify against the textbook
41. 41
    Q41 → answer not in source — verify against the textbook
42. 42
    Q42 → answer not in source — verify against the textbook
43. 43
    Q43 → LHD = −1 and RHD = 1 at x = 1; since they differ, f is not differentiable at x = 1.
44. 44
    Q44 → Not differentiable at x = 1 and x = 2: one-sided derivatives are infinite/0 and unequal.
45. 45
    Q45 → answer not in source — verify against the textbook
46. 46
    Q46 → answer not in source — verify against the textbook
47. 47
    Q47 → answer not in source — verify against the textbook
48. 48
    Q48 → answer not in source — verify against the textbook
49. 49
    Q49 → answer not in source — verify against the textbook
50. 50
    Q50 → answer not in source — verify against the textbook
51. 51
    Q51 → answer not in source — verify against the textbook
52. 52
    Q52 → answer not in source — verify against the textbook
53. 53
    Q53 → answer not in source — verify against the textbook
54. 54
    Q54 → answer not in source — verify against the textbook
55. 55
    Q55 → answer not in source — verify against the textbook
56. 56
    Q56 → answer not in source — verify against the textbook
57. 57
    Q57 → answer not in source — verify against the textbook
58. 58
    Q58 → answer not in source — verify against the textbook
59. 59
    Q59 → answer not in source — verify against the textbook
60. 60
    Q60 → answer not in source — verify against the textbook
61. 61
    Q61 → answer not in source — verify against the textbook
62. 62
    Q62 → answer not in source — verify against the textbook
63. 63
    Q63 → answer not in source — verify against the textbook
64. 64
    Q64 → answer not in source — verify against the textbook
65. 65
    Q65 → answer not in source — verify against the textbook
66. 66
    Q66 → answer not in source — verify against the textbook
67. 67
    Q67 → answer not in source — verify against the textbook
68. 68
    Q68 → answer not in source — verify against the textbook
69. 69
    Q69 → answer not in source — verify against the textbook
70. 70
    Q70 → answer not in source — verify against the textbook
71. 71
    Q71 → answer not in source — verify against the textbook
72. 72
    Q72 → answer not in source — verify against the textbook
73. 73
    Q73 → answer not in source — verify against the textbook
74. 74
    Q74 → answer not in source — verify against the textbook
75. 75
    Q75 → answer not in source — verify against the textbook
76. 76
    Q76 → answer not in source — verify against the textbook
77. 77
    Q77 → answer not in source — verify against the textbook
78. 78
    Q78 → answer not in source — verify against the textbook
79. 79
    Q79 → answer not in source — verify against the textbook
80. 80
    Q80 → answer not in source — verify against the textbook
81. 81
    Q81 → answer not in source — verify against the textbook
82. 82
    Q82 → answer not in source — verify against the textbook
83. 83
    Q83 → answer not in source — verify against the textbook
84. 84
    Q84 → answer not in source — verify against the textbook
85. 85
    Q85 → answer not in source — verify against the textbook
86. 86
    Q86 → answer not in source — verify against the textbook
87. 87
    Q87 → Both methods give u′vw + uv′w + uvw′, so the formula holds.
88. 88
    Q88 → answer not in source — verify against the textbook
89. 89
    Q89 → answer not in source — verify against the textbook
90. 90
    Q90 → answer not in source — verify against the textbook
91. 91
    Q91 → answer not in source — verify against the textbook
92. 92
    Q92 → answer not in source — verify against the textbook
93. 93
    Q93 → answer not in source — verify against the textbook
94. 94
    Q94 → answer not in source — verify against the textbook
95. 95
    Q95 → answer not in source — verify against the textbook
96. 96
    Q96 → answer not in source — verify against the textbook
97. 97
    Q97 → answer not in source — verify against the textbook
98. 98
    Q98 → Differentiating log x and log y and dividing gives dy/dx = −y/x.
99. 99
    Q99 → answer not in source — verify against the textbook
100. 100
     Q100 → answer not in source — verify against the textbook
101. 101
     Q101 → answer not in source — verify against the textbook
102. 102
     Q102 → answer not in source — verify against the textbook
103. 103
     Q103 → answer not in source — verify against the textbook
104. 104
     Q104 → answer not in source — verify against the textbook
105. 105
     Q105 → answer not in source — verify against the textbook
106. 106
     Q106 → answer not in source — verify against the textbook
107. 107
     Q107 → answer not in source — verify against the textbook
108. 108
     Q108 → answer not in source — verify against the textbook
109. 109
     Q109 → y″ = −5cos x + 3sin x = −y, so y″ + y = 0.
110. 110
     Q110 → answer not in source — verify against the textbook
111. 111
     Q111 → x y₁ = −3sin(log x) + 4cos(log x); differentiating and substituting gives x²y₂ + xy₁ + y = 0.
112. 112
     Q112 → y′ = mAe^{mx}+nBe^{nx}, y″ = m²Ae^{mx}+n²Be^{nx}; substitution using mn(Ae^{mx}+Be^{nx}) gives 0.
113. 113
     Q113 → y″ = 500·49 e^{7x} + 600·49 e^{−7x} = 49y.
114. 114
     Q114 → From e^y = 1/(x+1), y = −log(x+1), so y′ = −1/(x+1) and y″ = 1/(x+1)² = (y′)².
115. 115
     Q115 → y₁ = 2 tan⁻¹x/(1+x²); differentiating and clearing (1+x²)² gives the identity.
116. 116
     Q116 → f is continuous on [−4,2] and differentiable on (−4,2); f(−4) = f(2) = 0, so there is c with f′(c) = 0. f′(x) = 2x + 2 = 0 gives c = −1 ∈ (−4,2).
117. 117
     Q117 → Rolle's theorem applies to none. (i) and (ii): the greatest integer function is not continuous at the integers inside the interval. (iii): f is continuous and differentiable but f(1) = 0 ≠ f(2) = 3. Hence the converse of Rolle's theorem is not true.
118. 118
     Q118 → Suppose f(−5) = f(5); then by Rolle's theorem there is c ∈ (−5,5) with f′(c) = 0, contradicting the hypothesis. Hence f(−5) ≠ f(5).
119. 119
     Q119 → Slope (f(4)−f(1))/3 = 1; f′(c) = 2c − 4 = 1 gives c = 5/2 ∈ (1,4).
120. 120
     Q120 → answer not in source — verify against the textbook
121. 121
     Q121 → MVT applies only to (iii). (i) and (ii) are not continuous (nor differentiable) throughout the interval. For (iii), f′ obeys (f(2)−f(1))/1 = 3; f′(c) = 2c = 3 gives c = 3/2 ∈ (1,2).

MCQ Questions

1 Mark Questions

1. 1
   State the condition for f(x) to be continuous at x = a.
2. 2
   Write the derivative of eˣ.
3. 3
   Find dy/dx if y = x⁵ + 3x². (Ans: 5x⁴ + 6x)
4. 4
   What is the derivative of |x| at x = 0?

2 Mark Questions

1. 1
   Discuss continuity of f(x) = |x| at x = 0. (Ans: continuous, since left and right limits equal f(0) = 0)
2. 2
   Find dy/dx if y = (x² + 1)⁵. (Ans: 10x(x² + 1)⁴)
3. 3
   Check continuity of f(x) = (x² − 4)/(x − 2) at x = 2 after removing the removable discontinuity. (Ans: define f(2) = 4)

3 Mark Questions

1. 1
   If y = x^(sin x), find dy/dx using logarithmic differentiation. (Ans: x^(sin x)(cos x ln x + sin x/x))
2. 2
   Find dy/dx for x² + y² = 25 (implicit differentiation). (Ans: dy/dx = −x/y)
3. 3
   If x = a(t + sin t) and y = a(1 + cos t), find dy/dx at t = π/2. (Ans: −1)

4 Mark Questions

1. 1
   If y = e^(x sin x), prove that d²y/dx² − 2 dy/dx + (1 + sin x cos x) y = 0. Show each differentiation step.
2. 2
   Differentiate (sin x)^(cos x) with respect to x using logarithms, and also find d²y/dx² for y = x²/2 + x at x = 1. (Ans: first gives (sin x)^(cos x)[cos x cot x − sin x ln(sin x)]; second gives 1)