Verified Solution • 2025–26 Session moderate

QJEEM22Mad16b: JEE Main 2022 Mathematics - Class 12 JEEM22Mad16b Solution: Let f : R → R be defined as f ( x ) = [ [ e x ] , x < 0 a e x + …

Let f : R → R be defined as

f(x)=[[ex],x<0aex+[x−1],0≤x<1b+[sin⁡(πx)],1≤x<2[e−x]−c,x≥2

where a, b, c ∈ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

Multiple Choice Options

  • A There exists a, b, c ∈ R such that f is continuous on R.
  • B If f is discontinuous at exactly one point, then a + b + c = 1
  • C If f is discontinuous at exactly one point, then a + b + c ≠ 1
  • D f is discontinuous at at least two points, for any values of a, b and c

Step-by-Step Verified Solution

f(x)={0x<0aex−10≤x<1bx=1b−11<x<2−cx≥2

To be continuous at x = 0

a − 1 = 0

to be continuous at x = 1

ae − 1 = b = b − 1 ⇒ not possible

to be continuous at x = 2

b − 1 = − c ⇒ b + c = 1

If a = 1 and b + c = 1 then f(x) is discontinuous at exactly one point.

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