Let a function $f: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ccc}\sin x-e^{x} & \text { if } & x \leq 0 \\ a+[-x] & \text { if } & 0<x<1 \\ 2 x-b & \text { if } & x \geq 1\end{array}\right.$ where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then $(\mathrm{a}+\mathrm{b})$ is equal to :

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Continuity and Differentiability

Let a function

f : R \to R

be defined as

f (x) = ⎩ ⎨ ⎧ sin x - e^{x} a + [- x] 2 x - b if if if x \leq 0 0 < x < 1 x \geq 1

where

[x]

is the greatest integer less than or equal to

x

. If

f

is continuous on

R

, then

(a + b)

is equal to :

Answer: (C)

For

0 < x < 1

the number

- x

lies in

(- 1, 0)

, so

[- x] = - 1

and the middle branch is the constant

a - 1

. Continuity at

x = 0

matches this constant with

sin 0 - e^{0} = - 1

, so

a - 1 = - 1

and

a = 0

. Continuity at

x = 1

matches the same constant with

2 x - b

evaluated there, so

a - 1 = 2 - b \Rightarrow - 1 = 2 - b \Rightarrow b = 3

Adding the two values,

a + b = 3

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Single correct option