Question

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

• A. continuous if $a=-5$ and $b=10$
• B. continuous if $a=5$ and $b=5$
• 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 $b$

The given function will be continuous everywhere when it is continuous at $x=1,3\text{and}5$.

For continuity at $x=1$ ; $f\left(1^{-}\right)=f\left(1\right)=f\left(1^{+}\right)$
$a+b=5\cdots \left(i\right)$
Similarly, at $x=3$ and at $x=5$
$a+3b=b+15\cdots \left(ii\right)$
$b+25=30\cdots \left(iii\right)$
From $\left(i\right)$ and $\left(iii\right)$, we get $a=0,b=5$
From $\left(ii\right)$ and $\left(iii\right)$, we get $a=5,b=5$

Clearly, the function is not continuous for any values of $a$ and $b$.