Question:
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function defined as
$f(x)=\left\{\begin{array}{ccc}5, & \text { if } & x \leq 1 \\ \mathrm{a}+\mathrm{b} x, & \text { if } & 1 Then, $f$ is :
Correct Option: , 4
Solution:
Let $f(x)$ is continuous at $x=1$, then
$f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$\Rightarrow \quad 5=a+b$ ..........(1)
Let $f(x)$ is continuous at $x=3$, then
$f\left(3^{-}\right)=f(3)=f\left(3^{+}\right)$
$\Rightarrow \quad a+3 b=b+15$...........(2)
Let $f(x)$ is continuous at $x=5$, then
$f\left(5^{-}\right)=f(5)=f\left(5^{+}\right)$
$\Rightarrow \quad b+25=30$
$\Rightarrow b=30-25=5$
From (1), $a=0$
But $a=0, b=5$ do not satisfy equation (2)
Hence, $f(x)$ is not continuous for any values of $a$ and $b$