Question

Check if function is continuous or discontinuous

If $f\left(x\right)=\left\{\begin{array}{ll}5& ;x\le 1\\ a+bx& ;1<x<3\\ b+5x& ;3\le x<5\\ 30& ;x\ge 5\end{array}\right.$

• A. $f\left(x\right)$ is discontinuous $\forall a\in R,b\in R$
• B. $f\left(x\right)$ is continuous if $a=0\&b=5$
• C. $f\left(x\right)$ is continuous if $a=5\&b=0$
• D. $f\left(x\right)$ is continuous if $a=-5\&b=10$

Answer 1

The correct option is A

$f\left(x\right)$ is discontinuous $\forall a\in R,b\in R$

Explanation for the correct option:

Step 1. For continuity at $x=1$

$f\left(1\right)=\underset{h\to 0}{\lim }f(1+h)=\underset{h\to 0}{\lim }f(1-h)$

$f\left(1\right)=5$

$\underset{h\to 0}{\lim }f(1+h)=a+b$

$\underset{h\to 0}{\lim }f(1-h)=5$

$\Rightarrow$ $a+b=5$ ……(1)

Step 2. For continuity at $x=3$

$f\left(3\right)=\underset{h\to 0}{\lim }f(3+h)=\underset{h\to 0}{\lim }f(3-h)$

$f\left(3\right)=b+15$

$\underset{h\to 0}{\lim }f(3+h)=b+15$

$\underset{h\to 0}{\lim }f(3-h)=a+3b$

$\Rightarrow$ $a+2b=15$ ……(2)

Step 3. For continuity at $x=5$

$f\left(5\right)=\underset{h\to 0}{\lim }f(5+h)=\underset{h\to 0}{\lim }f(5-h)$

$\underset{h\to 0}{\lim }f(5-h)=b+25$

$\Rightarrow$ $b+25=30$

$\Rightarrow$ $b=5$ ……(3)

Now, from equation (1), (2) and (3), we get

$a=5,a=0$

$\therefore$ $f\left(x\right)$ is discontinuous $\forall a\in R,b\in R$

Hence, option ‘A’ is Correct.