Question

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

• A. Continuous if $a=5$ and $b=5$
• B. Continuous if $a=-5$ and $b=10$
• C. Continuous if $a=0$ and $b=5$
• D. Not continuous for any values of $a$ and $b$

Answer 1

The correct option is D Not continuous for any values of $a$ and $b$

$f\left(x\right)=\{\begin{array}{ccc}5,& if& x\le 1\\ a+bx,& if& 1<x<3\\ b+5x,& if& 3\le x<5\\ 30,& if& x\ge 5\end{array}$
$f\left(1\right)=5,f\left(1^{-}\right)=5,f\left(1^{+}\right)=a+b$
$f\left(3^{-}\right)=a+3b,f\left(3\right)=b+15,f\left(3^{+}\right)=b+15$
$f\left(5^{-}\right)=b+25:f\left(5\right)=30f\left(5^{+}\right)=30$
From above we concluded that $f$ is not continuous for any values of $a$ and $b$.