JEE Mathematics Continuity and Differentiability MCQs Set 02

Question. Let \( f(x) = \log |x - 1|, x \neq 1 \). The value of \( f' \left( \frac{1}{2} \right) \)
(a) is -2
(b) is 2
(c) does not exist
(d) None of the options
Answer: (a) is -2

Question. Let \( y = \left| \tan \left( \frac{\pi}{4} - x \right) \right| \). Then \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \)
(a) is 1
(b) is -1
(c) does not exist
(d) None of the options
Answer: (c) does not exist

Question. Let \( y = |x| + |x - 2| \). Then \( \frac{dy}{dx} \) at \( x = 2 \)
(a) is 2
(b) is 0
(c) does not exist
(d) None of the options
Answer: (c) does not exist

Question. Let \( f(x) = \lambda + \mu |x| + \nu |x|^2 \), where \( \lambda, \mu, \nu \) are real constants. The \( f'(0) \) exists if
(a) \( \mu = 0 \)
(b) \( \nu = 0 \)
(c) \( \lambda = 0 \)
(d) \( \mu = \nu \)
Answer: (a) \( \mu = 0 \)

Question. If \( f(x) = \frac{[x]}{|x|}, x \neq 0 \) where [.] denotes the greatest integer function, then \( f'(1) \) is
(a) -1
(b) \( \infty \)
(c) nonexistent
(d) None of the options
Answer: (c) nonexistent

Question. If \( f(x) = |\cos 2x| \) then \( f' \left( \frac{\pi}{4} + 0 \right) \) is equal to
(a) 2
(b) 0
(c) -2
(d) None of the options
Answer: (a) 2

Question. If \( f(x) = \sin \pi[x] \) then \( f'(1 - 0) \) is equal to
(a) -1
(b) 0
(c) 1
(d) None of the options
Answer: (b) 0

Question. Let \( f(x) = [x^2] - [x]^2 \), where [.] denotes the greatest integer function. Then
(a) \( f(x) \) is discontinuous for all integral values of x
(b) \( f(x) \) is discontinuous only at x = 0, 1
(c) \( f(x) \) is continuous only at x = 1
(d) None of the options
Answer: (c) \( f(x) \) is continuous only at x = 1

Question. Let \( f(x) = [\cos x + \sin x], 0 < x < 2\pi \) where [x] denotes the greatest integer less than or equal to x. The number of points of discontinuity of \( f(x) \) is
(a) 6
(b) 5
(c) 4
(d) 3
Answer: (c) 4

Question. Let \( f(x) = x - |x - x^2|, x \in [-1, 1] \). Then the number of points at which \( f(x) \) is discontinuous is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (c) 0

Question. Let \( f(x) = \begin{cases} \sqrt{1 + x^2}, & x < \sqrt{3} \\ \sqrt{3}x - 1, & \sqrt{3} \leq x < 4 \\ [x], & 4 \leq x < 5 \\ |1 - x|, & x \geq 5 \end{cases} \), where [x] is the greatest integer \( \leq x \). The number of points of discontinuity of \( f(x) \) in R is
(a) 3
(b) 0
(c) infinite
(d) None of the options
Answer: (d) None of the options

Question. Let \( f(x) = \int_0^x t \sin \frac{1}{t} dt \). Then the number of points of discontinuity of the function \( f(x) \) in the open interval \( (0, \pi) \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. Let \( f : [0, 1] \to [0, 1] \) be a continuous function. Then
(a) \( f(x) = x \) for at least one \( 0 \leq x \leq 1 \)
(b) \( f(x) \) will be differentiable in [0, 1]
(c) \( f(x) + x = 0 \) for at least one x such that \( 0 \leq x \leq 1 \)
(d) None of the options
Answer: (a) \( f(x) = x \) for at least one \( 0 \leq x \leq 1 \)

Question. Let \( f(x) \) be a continuous function defined for \( 1 \leq x \leq 3 \). If \( f(x) \) takes rational values for all x and \( f(2) = 10 \) then the value of \( f(1.5) \) is
(a) 7.5
(b) 10
(c) 5
(d) None of the options
Answer: (b) 10

Question. If \( f(x) = e^{-1/x^2}, x \neq 0 \), and \( f(0) = 0 \) then \( f'(0) \) is
(a) 0
(b) 1
(c) e
(d) nonexistent
Answer: (c) e

Question. Let \( f(x) = \sin x, g(x) = [x + 1] \) and \( g\{f(x)\} = h(x) \), where [.] is the greatest integer function. Then \( h' \left( \frac{\pi}{2} \right) \) is
(a) nonexistent
(b) 1
(c) -1
(d) None of the options
Answer: (a) nonexistent

Question. Let \( f(x) = [x], g(x) = |x| \) and \( f\{g(x)\} = h(x) \), where [.] is the greatest integer function. Then \( h'(-1) \) is
(a) 0
(b) \( -\infty \)
(c) nonexistent
(d) None of the options
Answer: (c) nonexistent

Question. The number of values of \( x \in [0, 2] \) at which the real function \( f(x) = \left| x - \frac{1}{2} \right| + |x - 1| + \tan x \) is not finitely differentiable is
(a) 2
(b) 3
(c) 1
(d) 0
Answer: (b) 3

Question. Let \( f(x) = [n + p \sin x], x \in (0, \pi), n \in \mathbb{Z} \), p is a prime number and [x] = the greatest integer less than or equal to x. The number of points at which \( f(x) \) is not differentiable is
(a) p
(b) p – 1
(c) 2p + 1
(d) 2p – 1
Answer: (d) 2p – 1

Question. Let \( f(x) = \begin{cases} (x - 1)^2 \cos \frac{1}{x - 1} - |x|, & x \neq 1 \\ -1, & x = 1 \end{cases} \). The set of points where \( f(x) \) is not differentiable is
(a) {1}
(b) {0, 1}
(c) {0}
(d) None of the options
Answer: (c) {0}

Question. Let \( f(x) = \frac{1 - \sin x}{\sin 2x}, x \neq \frac{\pi}{2} \). If \( f(x) \) is continuous at \( x = \frac{\pi}{2} \) then \( f \left( \frac{\pi}{2} \right) \) should be
(a) 1
(b) 0
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (b) 0

Question. A function \( f(x) \) is defined as below: \( f(x) = \frac{\cos(\sin x) - \cos x}{x^2}, x \neq 0 \) and \( f(0) = a \). \( f(x) \) is continuous at x = 0 if a equals
(a) 0
(b) 4
(c) 5
(d) 6
Answer: (a) 0

Question. Let \( f(x) = \begin{cases} \frac{1 - \tan x}{4x - \pi}, & x \neq \frac{\pi}{4} \text{ and } x \in [0, \frac{\pi}{2}) \\ \lambda, & x = \frac{\pi}{4} \end{cases} \). If \( f(x) \) is continuous in \( [0, \frac{\pi}{2}) \) then \( \lambda \) is
(a) 1
(b) \( \frac{1}{2} \)
(c) \( -\frac{1}{2} \)
(d) None of the options
Answer: (c) \( -\frac{1}{2} \)

Question. Let \( f(x) = (\sin x)^{\frac{1}{\pi - 2x}}, x \neq \frac{\pi}{2} \). If \( f(x) \) is continuous at \( x = \frac{\pi}{2} \) then \( f \left( \frac{\pi}{2} \right) \) is
(a) e
(b) 1
(c) 0
(d) None of the options
Answer: (b) 1

Question. Let \( f(x) = \sin \frac{1}{x}, x \neq 0 \). Then \( f(x) \) can be continuous at x = 0
(a) if f(0) = 1
(b) if f(0) = 0
(c) if f(0) = -1
(d) for no value of f(0)
Answer: (d) for no value of f(0)

Question. If \( f(x) = \begin{cases} px^2 - q, & x \in [0, 1) \\ x + 1, & x \in (1, 2] \end{cases} \) and \( f(1) = 2 \) then the value of the pair (p, q) for which \( f(x) \) cannot be continuous at x = 1 is
(a) (2, 0)
(b) (1, -1)
(c) (4, 2)
(d) (1, 1)
Answer: (d) (1, 1)

Question. If \( f(x) = x, x \leq 1 \), and \( f(x) = x^2 + bx + c, x > 1 \), and \( f'(x) \) exists finitely for all \( x \in \mathbb{R} \) then
(a) b = -1, c \( \in \mathbb{R} \)
(b) c = 1, b \( \in \mathbb{R} \)
(c) b = 1, c = -1
(d) b = -1, c = 1
Answer: (d) b = -1, c = 1

Choose the correct options. One or more options may be correct.

Question. If \( f(x) = \begin{cases} e^x, & x < 2 \\ a + bx, & x \geq 2 \end{cases} \) is differentiable for all \( x \in \mathbb{R} \) then
(a) a + b = 0
(b) a + 2b = e^2
(c) b = e^2
(d) None of the options
Answer: (a) a + b = 0, (b) a + 2b = e^2, (c) b = e^2

Question. If \( f(x) = \cos^{-1}(\cos x) \) then \( f(x) \) is
(a) continuous at \( x = \pi \)
(b) discontinuous \( x = -\pi \)
(c) differentiable at \( x = 0 \)
(d) nondifferentiable at \( x = \pi \)
Answer: (a) continuous at \( x = \pi \), (d) nondifferentiable at \( x = \pi \)

Question. Let \( f(x) = x - |x| \). Then
(a) \( f(x) \) is continuous everywhere
(b) \( f(x) \) is differentiable everywhere
(c) \( f(x) \) is discontinuous at x = 0
(d) \( f(x) \) is not differentiable at x = 0
Answer: (a) \( f(x) \) is continuous everywhere, (d) \( f(x) \) is not differentiable at x = 0

Question. If \( f(x) = [x] + \left[ x + \frac{1}{2} \right] \), where [.] denotes the greatest integer function, then
(a) \( f(x) \) is continuous at \( x = \frac{1}{2} \)
(b) \( \lim_{x \to 1/2+0} f(x) = 1 \)
(c) \( f(x) \) is discontinuous at \( x = \frac{1}{2} \)
(d) \( \lim_{x \to 1/2-0} f(x) = 1 \)
Answer: (a) \( f(x) \) is continuous at \( x = \frac{1}{2} \), (b) \( \lim_{x \to 1/2+0} f(x) = 1 \)

Question. If \( f(x) = |2 - x| + (2 + x) \), where (x) = the least integer greater than or equal to x, then
(a) f(2 – 0) = f(2) = 4
(b) \( f(x) \) is continuous at x = 2
(c) \( f(x) \) is nondifferentiable at x= 2
(d) \( f(x) \) is differentiable but not continuous at x = 2
Answer: (a) f(2 – 0) = f(2) = 4, (c) \( f(x) \) is nondifferentiable at x= 2

Question. Let \( h(x) = \min \{x, x^2\} \) for every real number x. Then
(a) h is continuous for all x
(b) h is differentiable for all x
(c) \( h'(x) = 1 \) for all x > 1
(d) h is not differentiable at two values of x
Answer: (a) h is continuous for all x, (c) \( h'(x) = 1 \) for all x > 1, (d) h is not differentiable at two values of x

Question. At x = 0, the function \( y = e^{-|x|} \) is
(a) continuous
(b) continuous and differentiable
(c) differentiable with derivative = 1
(d) differentiable with derivative = -1
Answer: (a) continuous

Question. A function \( f(x) \) is defined as follows : \( f(x) = -x^2, x \leq 0 \); \( f(x) = 5x – 4, 0 < x \leq 1 \); \( f(x) = 4x^2 – 3x, 1 < x \leq 2 \); \( f(x) = 3x + 4, x > 2 \).
(a) \( f(x) \) is not continuous at x = 0, but differentiable there
(b) \( f(x) \) is continuous at x = 1, but not differentiable there
(c) \( f(x) \) is continuous at x = 2, but not differentiable there
(d) None of the options
Answer: (c) \( f(x) \) is continuous at x = 2, but not differentiable there

Question. The function \( f(x) = \frac{1}{x} - \frac{2}{e^{2x} - 1}, x \neq 0 \), is continuous at x = 0. Then
(a) f(0) = 1
(b) \( f(x) \) is differentiable at x = 0
(c) \( f(x) \) is not differentiable at x = 0
(d) \( f'(0) = \frac{1}{3} \)
Answer: (a) f(0) = 1, (b) \( f(x) \) is differentiable at x = 0

Question. The function \( f(x) = |x^2 – 3x + 2| + \cos |x| \) is not differentiable at x =
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1, (d) 2

Question. Let \( f(x) \) be defined as follows : \( f(x) = x^6, x^2 > 1 \) and \( f(x) = x^3, x^2 \leq 1 \). Then \( f(x) \) is
(a) continuous everywhere
(b) differentiable everywhere
(c) discontinuous at x = -1
(d) not differentiable at x = 1
Answer: (c) discontinuous at x = -1, (d) not differentiable at x = 1

Question. Let \( f(x) = \begin{cases} \sin x, & x \geq 0 \\ -\sin x, & x < 0 \end{cases} \). The \( f(x) \) is
(a) continuous at x = 0
(b) differentiable at x = 0
(c) discontinuous at x = 0
(d) not differentiable at x = 0
Answer: (a) continuous at x = 0, (d) not differentiable at x = 0

Question. If \( f(x) = \sum_{n=0}^n a_n |x|^n \), where \( a_i \)’s are real constants, then \( f(x) \) is
(a) continuous at x = 0 for all \( a_i \)
(b) differentiable at x = 0 for all \( a_i \in \mathbb{R} \)
(c) differentiable at x = 0 for \( a_{2k+1} = 0 \)
(d) None of the options
Answer: (a) continuous at x = 0 for all \( a_i \), (c) differentiable at x = 0 for \( a_{2k+1} = 0 \)

Question. Let [x] denote the greatest integer less than or equal to x. Now g(x) is defined as below : \( g(x) = [f(x)], x \in \left( 0, \frac{\pi}{2} \right) \cup \left( \frac{\pi}{2}, \pi \right) \) and \( g(x) = 3, x = \frac{\pi}{2} \), where \( f(x) = \frac{2(\sin x - \sin^n x) + |\sin x - \sin^n x|}{2(\sin x - \sin^n x) - |\sin x - \sin^n x|}, n \in \mathbb{R} \). Then
(a) g(x) is continuous and differentiable at \( x = \frac{\pi}{2} \) when n > 1
(b) g(x) is continuous and differentiable at \( x = \frac{\pi}{2} \) when 0 < n < 1
(c) g(x) is continuous but not differentiable at \( x = \frac{\pi}{2} \) when n > 1
(d) g(x) is continuous but differentiable at \( x = \frac{\pi}{2} \) when 0 < n < 1
Answer: (b) g(x) is continuous and differentiable at \( x = \frac{\pi}{2} \) when 0 < n < 1

Question. Let \( f(x) = \phi(x) + \psi(x) \) and \( \phi'(a), \psi'(a) \) are finite and definite. Then
(a) \( f(x) \) is continuous at x = a
(b) \( f(x) \) is differentiable at x = a
(c) \( f(x) \) is continuous at x = a
(d) \( f'(x) \) is differentiable at x = a
Answer: (a) \( f(x) \) is continuous at x = a, (b) \( f(x) \) is differentiable at x = a

Question. Let \( f(x) = x + |x| \). Then \( f(x) \) is
(a) differentiable at all x
(b) continuous at all x
(c) differentiable everywhere except at x = 0
(d) continuous everywhere except at x = 0
Answer: (b) continuous at all x, (c) differentiable everywhere except at x = 0

Question. Let \( f(x) = \lim_{n \to \infty} \frac{1 - x^n}{1 + x^n} \). Then
(a) \( f(x) \) is a constant in 0 < x < 1
(b) \( f(x) \) is continuous at x = 1
(c) \( f(x) \) is not differentiable at x = 1
(d) None of the options
Answer: (a) \( f(x) \) is a constant in 0 < x < 1, (c) \( f(x) \) is not differentiable at x = 1

Question. Let \( f(x) = 1 - |\cos x| \) for all \( x \in \mathbb{R} \). Then
(a) \( f' \left( \frac{\pi}{2} \right) \) does not exist
(b) \( f(x) \) is continuous everywhere
(c) \( f(x) \) is not differentiable anywhere
(d) \( \lim_{x \to \pi/2 + 0} f(x) = 1 \)
Answer: (b) \( f(x) \) is continuous everywhere, (d) \( \lim_{x \to \pi/2 + 0} f(x) = 1 \)

Question. Let \( f(x) = [\tan^2 x] \), where [,] denotes the greatest integer function. Then
(a) \( \lim_{x \to 0} f(x) \) does not exist
(b) \( f(x) \) is continuous at x = 0
(c) \( f'(0) = 1 \)
(d) \( f(x) \) is not differentiable at x = 0
Answer: (b) \( f(x) \) is continuous at x = 0

Question. If \( f(x) = \frac{x}{\sqrt{x+1} - \sqrt{x}} \) be a real-valued function then
(a) \( f(x) \) is continuous, but \( f'(0) \) does not exist
(b) \( f(x) \) is differentiable at x = 0
(c) \( f(x) \) is not continuous at x = 0
(d) \( f(x) \) is not differentiable at x = 0
Answer: (b) \( f(x) \) is differentiable at x = 0

Question. A function \( f(x) \) is defined in the interval [1, 4) as follows : \( f(x) = \log_e[x], 1 \leq x < 3 \) and \( |\log_e x|, 3 \leq x < 4 \). The graph of the function \( f(x) \)
(a) is broken at two points
(b) is broken at exactly one point
(c) does not have a definite tangent at two points
(d) does not have a definite tangent at more than two points
Answer: (a) is broken at two points, (c) does not have a definite tangent at two points