If f(x) = { (sin⁻¹x)² cos(1/x), x ≠ 0; 0, x = 0 }, then

f(x) is continuous everywhere in x ∈ (−1,1)
f(x) is discontinuous in x ∈ [−1,1]
f(x) is differentiable everywhere in x ∈ (−1,1)
f(x) is non-differentiable anywhere in x ∈ [−1,1]

Correct answer: f(x) is continuous everywhere in x ∈ (−1,1)

Solution

The function f(x) = { (sin⁻¹x)² cos(1/x), x ≠ 0; 0, x = 0 } is continuous everywhere in x ∈ (−1,1) because sin⁻¹x is continuous in the interval [-1,1] and (sin⁻¹x)² is also continuous, and cos(1/x) is continuous for x ≠ 0, and at x = 0, the limit of f(x) as x approaches 0 is equal to 0, so f(x) is continuous at x = 0

Related JEE Advanced Maths questions

For the function f(x) = sin(x) + cos(x) defined on the interval [0, 2π], which of the following is true about its behavior?
Consider a function f defined on the interval [0, 2] such that it is continuous on [0, 2] and differentiable on (0, 2), with f(0) = 1. Define F(x) = ∫₀ˣ f(√t) dt for x ∈ [0, 2]. If it is given that F'(x) = f'(x) for every x in (0, 2), what is the value of F(2)?
Consider two continuous functions f and g defined on the interval [-1, 2], which are also twice differentiable on (-1, 2). The values of f and g at x = -1, 0, and 2 are provided in the table below: x = -1 x = 0 x = 2 f(x) = 3 f(x) = 6 f(x) = 0 g(x) = 0 g(x) = 1 g(x) = -1 In the intervals (-1, 0) and (0, 2), the second derivative of (f - 3g), denoted as (f - 3g)'', does not become zero at any point. Which of the following statements is true?
Suppose f: R → (0, 1) is a continuous function. Which of the following functions equals zero at least at one point within the interval (0, 1)?
Consider the function f(x) = x + ln(x) − x ln(x), where x lies in the interval (0, ∞). - Column 1 contains details about the zeros of f(x), f'(x), and f''(x). - Column 2 contains information about the behavior of f(x), f'(x), and f''(x) as x approaches infinity. - Column 3 contains details about the increasing or decreasing nature of f(x) and f'(x). Column 1: (I) f(x) = 0 for some x in (1, e²) (II) f'(x) = 0 for some x in (1, e) (III) f'(x) = 0 for some x in (0, 1) (IV) f''(x) = 0 for some x in (1, e) Column 2: (i) lim x→∞ f(x) = 0 (ii) lim x→∞ f(x) = −∞ (iii) lim x→∞ f'(x) = −∞ (iv) lim x→∞ f''(x) = 0 Column 3: (P) f is increasing on (0, 1) (Q) f is decreasing on (e, e²) (R) f' is increasing on (0, 1) (S) f' is decreasing on (e, e²) Which of the following options gives the correct combination?
Let f: R → R be a function that is twice differentiable, satisfying f''(x) > 0 for every x in R, with f(1/2) = 1/2 and f(1) = 1. Which of the following is true about f'(1)?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →