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Solved assignment on continuity
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1.Question: , then correct statement is โ Options: (a) (b) f(x) is continuous at x = 3 (c) f(x) is continuous at x = 1 (d) f(x) is continuous at x = 1 and 3 Answer: (c) Solution: At x=1, LHL = 3, RHL = 3 & f(1)= & continuous At x=3, LHL=11, RHL= 2.Question: Options: (a) (b) (c) f(x) is discontinuous at x = 0 (d) f(x) is continuous Answer: (c) Solution: 3.Question: If function f(x) , is continuous function, then f(0) is equal to - Options:
(a) 2 (b) 1/ (c) 1/ (d) 1/ Answer: (c) Solution: 4.Question: is continuous at x= 2, then a is equal to - Options: (a) 0 (b) 1 (c) - (d) 2 Answer: (a) Solution: 2-a = 2 a= 0 5.Question: is continuous at x = 0, then k is equal to - Options: (a) 2a + b (b) 2a โ b (c) b โ 2a (d) a + b Answer: (a) Solution:
8.Question: Function f(x) is discontinuous at - Options: (a) One point (b) Two points (c) Three points (d) Infinite number of points Answer: (c) Solution: where log is not defined. 9.Question: Which of the following functions has finite number of points of discontinuity in R(where [.] denotes greatest integer) Options: (A) tan x (B) |x| / x (C) x + [x] (D) sin [๏ฐ x] Answer: (b) Solution: tan x has infinite points of discontinuity at x=(2n+1) 2 are discontinuous at many integers. 10.Question: is a continuous functions, then f(๏ฐ/4) is equal to - Options: (a) โ1/
(b) 1/ (c) 1 (d) โ Answer: (a) Solution: 11.Question: The value of f(0), so that function, f(x) = becomes continuous for all x, is given by: Options: (a) (b) (c) (d) Answer: (b) Solution: Rationalize Numerator & denominator both 12.Question: If f(x) is continuous at x = 0, then - Options: (a) (b) [f(0)] = โ (c) {f(0)} = โ0. (d) [f(0)]. {f(0)}= โ1. where [x] and {x} denotes greatest integer and fractional part function. Answer: (d) Solution:
Answer: (b) Solution: SELECT THE CORRECT ALTERNATIVES (ONE OR MORE THAN ONE CORRECT ANSWERS 15.Question: The value(s) of x for which is continuous, is (are) - Options: (a) 3 (b) โ (d) 5 (d) Answer: (a), (b) Solution: is discontinuous when 16.Question: Which of the following function(s) not defined at x = 0 has/have removable discontinuity at the origin? Options: (a) (b) (c) (d) Answer: (b), (c), (d)
Solution: (a) Non removable discontinuity (b) (c) (d) 17.Question: Function whose jump (non-negative difference of LHL & RHL) of discontinuity is greater than or equal to one, is/are - Options: (a) (b) (c) (d) Answer: (a),(c),(d) Solution: (a) (b)
Unit exist but discontinuous does not exist ; Let check continuity at integers of go f ( x ) : fog ( x ) :
1.Question: Consider the piecewise defined function choose the answer which best describes the continuity of this function - Options: (a) the function is unbounded and therefore cannot be continuous (b) the function is right continuous at x = 0 (c) the function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line (d) the function is continuous on the entire real line Answer: (b),(d) Solution: It is continuous everywhere & so right continuous too at x = 0. 2.Question: f(x) is continuous at x=0, then which of the following are always true? Options: (a) (b) f(x) is non continuous at x= (c) g(x) = x^2 f(x) is continuous at x = 0 (d) Answer: (c),(d) Solution: If f(x) is continuous at x = 0 then x^2 f(x) is also continuous at x= & both are equal. 3.Question: Indicate all correct alternatives if, , then on the interval [0,๏ฐ] Options: (a) tan (f (x)) & are both continuous
Answer: (a),(c),(d) Solution: 6.Question: The number of points where f(x) = [sinx + cosx] (where [ ] denotes the greatest integer function), x ๏(0, 2๏ฐ) is not continuous is - Options: (a) 3 (b) 4 (c) 5 (d) 6 Answer: (c) Solution: 7.Question: On the interval I = [โ2, 2], the function Options: (a) is continuous for all values of x ๏ I (b) is continuous for x ๏ I โ(0) (c) assumes all intermediate values from f(โ2) & f(2) (d) has a maximum value equal to 3/e Answer: (b),(c),(d) Solution:
8.Question: If ; where [x] is the greatest integer function of x, then f(x) is continuous at - Options: (A) x = 0 (B) x = 1 (C) x = 2 (D) none of these Answer: (b),(c) Solution: 9.Question: Given where { } & [ ] denotes the fractional part and the integral part functions respectively, then which of the following statement does not hold good - Options: (a) f (0โ^ ) = 0 (b) f(0+^ )= (c) f(0)=0 continuity of f at x = 0 (d) irremovable discontinuity of f at x = 0 Answer: (b),(d)
12.Question: Consider where [] & {} are the greatest integer function & fractional part function respectively, then - Options: (a) f(0) = l n2 f is continuous at x = 0 (b) f(0) = 2 ๏ f is continuous at x = 0 (c) f(0) = e^2 ๏ ๏ f is continuous at x = 0 (d) f has an irremovable discontinuity at x = 0 Answer: (d) Solution: 13.Question: Let then - Options: (a) (b)
(c) (d) f is continuous at x = 0 Answer: (a) Solution: 14.Question: Consider Options: (A) f is continuous at x = 1 (B) f has a finite discontinuity at x = 1 (C) f has an infinite or oscillatory discontinuity at x = 1 (D) f has a removable type of discontinuity at x= Answer: (b) Solution:
