Let f: R → R be defined as f(x) = [ [eˣ], x < 0 a eˣ + [x - 1], 0 ≤ x < 1 b + [ sin (π x)], 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?

(C) If f is discontinuous at exactly one point, then a + b + c ≠ 1

(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

(D) f is discontinuous at at least two points, for any values of a, b and c

Limit, Continuity and Differentiability: JEE Main 2022 Mathematics question

JEE Main 2022 · 28 Jun, Morning shift · Q4152

JEE Main 202228 Jun, Morning shiftMathematicsMediumLimit, Continuity and Differentiability420× since 2002

Let f : R

\to

R be defined as

f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.

where a, b, c

\in

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