Question

If $f\left(x\right)=\{\begin{array}{cc}|x|-3,& x<1\\ |x-2|+a,& x\ge 1\end{array}$ and

$g\left(x\right)=\{\begin{array}{cc}2-|x|,& x<2\\ \text{sgn}\left(x\right)-b,& x\ge 2\end{array}$,

where $\text{sgn}\left(x\right)$ denotes the signum function. If $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is discontinuous at exactly one point, then which of the following values of $a$ and $b$ are possible?

• A. $a=-3,b=0$
• B. $a=2,b=1$
• C. $a=2,b=0$
• D. $a=-3,b=1$

Answer 1

Answer: (A), (B)

The correct options are
A $a=-3,b=0$
B $a=2,b=1$

$f\left(x\right)=\{\begin{array}{cc}|x|-3,& x<1\\ |x-2|+a,& x\ge 1\end{array}$

$f\left(x\right)$ is continuous for all $x$ if it is continuous at $x=1$, for which $\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}f\left(x\right)=f\left(1\right)$
$\Rightarrow |1|-3=|1-2|+a$
$\Rightarrow a=-3$

$g\left(x\right)=\{\begin{array}{cc}2-|x|,& x<2\\ \text{sgn}\left(x\right)-b,& x\ge 2\end{array}$

$g\left(x\right)$ is continuous for all $x$ if it is continuous at $x=2$, for which $\underset{x\to 2^{-}}{lim}g\left(x\right)=\underset{x\to 2^{+}}{lim}g\left(x\right)=g\left(2\right)$
$\Rightarrow 2-|2|=\text{sgn}\left(2\right)-b=1-b$
$\Rightarrow b=1$

Thus, $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is continuous for all $x$ if $a=-3,b=1$
Hence, $h\left(x\right)$ is discontinuous at exactly one point for $(a=-3,b=0)$ and $(a=2,b=1)$