Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined as $f(x)=\left\{\begin{array}{ccc}5, & \text { if } & x \leq 1 \\ a+bx, & \text { if } & 1<x<3 \\ b+5x, & \text { if } & 3 \leq x<5 \\ 30, & \text { if } & x \geq 5\end{array}\right.$. Then $f$ is:

(D) not continuous for any values of a and b

(A) continuous if $a=5$ and $b=5$

(B) continuous if $a=5$ and $b=10$

(C) continuous if $a=0$ and $b=5$

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Continuity and Differentiability

Let

f : R \to R

be a function defined as

f (x) = ⎩ ⎨ ⎧ 5, a + b x, b + 5 x, 30, if if if if x \leq 1 1 < x < 3 3 \leq x < 5 x \geq 5

. Then

f

is:

Answer: (D)

Check continuity at each join. At

x = 1

, matching the left value

5

with

a + b x

needs

a + b = 5

. At

x = 3

, matching

a + 3 b

with

b + 15

needs

a + 2 b = 15

. At

x = 5

, matching

b + 25

with

30

needs

b = 5

, and then

a = 5

from

a + 2 b = 15

. But those values give

a + b = 10

, not

5

, so the requirement at

x = 1

breaks. No choice of

a

and

b

can make

f

continuous.

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option