Revision Notes: Continuity and Differentiability | Mathematics (Maths) for JEE Main & Advanced PDF Download

Important Formulas

1. Let f and g be two real functions with a common domain D, then.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

2. Expansions of Some Functions

(i)

(ii)(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

3. L’Hôpital’s rule

For functions f and g which are differentiable:has a finite value then

4. Common Indeterminate Forms

5. Differentiation

6. Continuity

Function f(x) is continuous at x = a if

7. Differentiability
f(x) is said to be derivable or differentiable at x = a if f’(a+) = f’(a–) = finite quantity

8. Rolle’s theorem
If a function f defined on the closed interval [a, b] is continuous on [a, b] and derivable on (a, b) and f(a) = f(b), then there exists at least one real number c between a and b (a < c < b), such that f ‘ (c) = 0

9. Mean Value Theorem
If a function f defined on the closed interval [a, b], is continuous on [a, b] and derivable on (a, b), then there exists at least one real number c between a and b (a < c < b), such that

Solved Examples

Que1: The function
Determine the values of 'a' & 'b', if f is continuous at x = π/2

Ans:

⇒ b + 2 = 1
b = – 1
(as sin h > 0, for h → 0⁺ and cot h > 0, for h → 0⁺) 
Now, this limit will be of the form equation for non zero values of a for a = 0,

Que2: LetIf f(x) is continuous at x = 0, then find the value of (a² + b²).

Ans: At x = 0

Que3: If

Examine the continuity of f(x) at x = 1.

Ans: In order to examine the continuity at x = 1, we are required to derive the definition of f(x) in the intervals x < 1, x > 1, and at x = 1, i.e., on and around x = 1.
Now, if 0 < x < 1,

Thus, we have

So, f(x) is not continuous at x = 1.

Que4: Which of the following functions defined below are NOT differentiable at the indicated point?
(a) (b)

(c) (d)

Ans: (d) A for f(x), at x = 0,
: LHD = RHD, f(x) is differentiable

(b) for g(x), at x = 0
: LHD = RHD, g(x) is differentiable

(c) for h(x) at x = 0
: LHD = RHD, h(x) is differentiable
(d) for k(x), at x = 1
: LHD ≠ RHD, k(x) is not differentiable at x = 1

Que5:If f(x) is derivable ∀ x ∈ R, then
(a) 2a + bπ = 7
(b) b + 2π = 3
(c) 2a + bπ = 13
(d) None of these

Ans: (d) f(x) must also be continuous at x = 2
∴ LHL = RHL = Vf(x) = 2
for derivability,
LHD = RHD
(2x + 2) |_x=2 = (a cos(πx)) |_x=2 ⇒ 6 = a
∴ b = 11, a = 6
∴2a + bπ = 12 + 11π ≠ 7
b + 2π = 11 + 2π ≠ 3
2a + bπ = 12 + 11π ≠ 13
No correct option.

Que6: [x] denotes the greatest integer less than or equal to x. If f(x) = [sin(πx)] in (-1, 1) then f(x) is -
(a) Continuous at x = 0
(b) Continuous in (-1, 0) ∪ (0, 1)
(c) Differentiable in (-1, 1)
(d) None

Ans: (b) Function can be discontinuous at x = 0
At x = 0
As x → 0⁻, [x] → -1, [sin(πx)] → -1
∴ LHL = -1 × -1 = 1
As x → 0⁺, [x] → 0, [sin(πx)] → 0
∴ RHL = 0
: LHL ≠ RHL, f(x) is discontinuous at x = 0.
It can also be discontinuous at x = 1/2.
At x = 1/2
continuous at x = -1/2 (∴ LHL = RHL)
At x = -1/2,
∴ [sin(πx)] = -1
∴ LHL = -1 × -1 = 1
∴ RHL = -1 × -1 = 1
∴ LHL = RHL, continuous at x = -1/2
f(x) is continuous in x ∈ (-1, 0) ∪ (0, 1)

Que7: The function f(x) = sin |x| is
(a) Continuous for all x
(b) Continuous only at certain points
(c) Differentiable at all points
(d) None of these

Ans:

for h → 0⁺
Now, |-h| = h, [h] = 0,
sin(-h) = –sin h < 0 ∀ sgn |sin(-h) ) | = –1

Que8:  Function

(a) Continuous everywhere
(b) Differentiable everywhere
(c) Differentiable everywhere except at x = 0
(d) None of these

Ans: (c)

∴ Function is discontinuous and not differentiable at x = 0

Que9: If 

then the function f(x) is differentiable for -
(a) x ∈ R⁺
(b) x ∈ R
(c) x ∈ R₀
(d) None of these

Ans: (c)

∴ Function is not differentiable at x = 0

Que10: If 

is differentiable at x = 0, then -
(a) α > 0
(b) α > 1
(c) α ≥ 1
(d) α ≥ 0

Ans: (b)

∴ for function to be differentiable
f'(0⁻) = f'(0⁺)
⇒ (-h)⁰⁻¹ = (h)⁰⁻¹
⇒ (-1)⁰⁻¹(h)⁰⁻¹ = (h)⁰⁻¹
∴ α = even ⇒ α > 1