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Continuity and Differentiability · Question 1

JEE Advanced 2026 Paper 1Multiple CorrectMedium📈 Topic in 33% papers+4−1
1

Let R denote the set of all real numbers. Let f : R R be an arbitrary function and let g : R R be the function defined by g(x) = xf(x) , for all x R . Then which of the following statements is (are) TRUE ?

  1. The function g is always continuous at x = 0
  2. If f is continuous at x = 0 , then g is differentiable at x = 0
  3. If g is differentiable at x = 0 , then f is continuous at x = 0
  4. If g is differentiable at x = 0 , then x 0 f(x) exists

Answer

(2) If f is continuous at x = 0 , then g is differentiable at x = 0, (4) If g is differentiable at x = 0 , then x 0 f(x) exists

Solution

Key: g(0) = 0, so g(x) - g(0)x - 0 = f(x) for x 0; hence g'(0) exists x 0 f(x) exists.

(A) false: f(x) = 1x (x 0), f(0) = 0 gives g(x) = 1 for x 0, g(0) = 0; discontinuous at 0.

(B) true: if f is continuous at 0, g'(0) = x 0 f(x) = f(0).

(C) false: f(x) = 0 (x 0), f(0) = 1 gives g 0, differentiable, but f is discontinuous at 0.

(D) true: g'(0) = x 0 f(x), so the limit exists.

Answer: (B), (D)

JEE Advanced 2026 Paper 1

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2

Consider the function f : (- 2, 2 ) (- , ) defined by f(x) = (|x| + |x - 1|) x + [x x] , where [x x] is the greatest integer less than or equal to x x . Let be the total number of points in the interval (- 2, 2 ) at which f is NOT continuous, and let be the total number of points in the interval (- 2, 2 ) at which f is NOT differentiable. Then the value of + is .

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3

For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S, let |S| denote the number of elements in the set S. Consider the functions f:(-3,3) (- , ) and g:(-3,3) (- , ) defined by f(x)=[x3] e (1+ 2 ( (x-[x]) ) ) and g(x)=x3 2 ( e(1+x-[x]) ). Let A=\x (-3,3): f is discontinuous at x\ and B=\x (-3,3): g is discontinuous at x\. Then the value of |A|+2|B|-|A B| is .

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4

Let R denote the set of all real numbers. Let f: R R and g: R (0,4) be functions defined by f(x)= e(x2+2x+4), and g(x)= 41+e-2x. Define the composite function f g-1 by (f g-1)(x)=f (g-1(x) ), where g-1 is the inverse of the function g. Then the value of the derivative of the composite function f g-1 at x=2 is .

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5

Let f: R R and g: R R be functions defined by f(x)= cases x|x| ( 1x ), & x 0, \ 0, & x=0, cases and g(x)= cases 1-2x, & 0 x 12, \ 0, & otherwise. cases Let a,b,c,d R. Define the function h: R R by h(x)=a f(x)+b (g(x)+g ( 12-x ) )+c (x-g(x) )+d g(x), x R. Match each entry in List-I to the correct entry in List-II.

  1. (P) → (4) (Q) → (3) (R) → (1) (S) → (2)
  2. (P) → (5) (Q) → (2) (R) → (4) (S) → (3)
  3. (P) → (5) (Q) → (3) (R) → (2) (S) → (4)
  4. (P) → (4) (Q) → (2) (R) → (1) (S) → (3)
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