Question

$f$ is a continuous function in $[a,b]$ and $g$ is a continuous function $h\left(x\right)$ is defined as
$h\left(x\right)=\{\begin{array}{c}f\left(x\right),x\in [a,b)\\ g\left(x\right),x\in (b,c]\end{array}$
Now, if $f\left(b\right)=g\left(b\right)$, then

• A. $h\left(x\right)$ may or may not be continuous in $[a,c]$
• B. $h\left(b^{+}\right)=g\left(b^{-}\right)$ and $h\left(b^{-}\right)=f\left(b^{+}\right)$
• C. $h\left(b^{-}\right)=g\left(b^{+}\right)$ and $h\left(b^{+}\right)=f\left(b^{-}\right)$
• D. $h\left(x\right)$ has a removable discontinuity at $x=b$

Answer 1

The correct option is A $h\left(x\right)$ may or may not be continuous in $[a,c]$

At $x=b$

${lim}_{x\to b^{-}}h\left(x\right)={lim}_{x\to b^{-}}f\left(x\right)=f\left(b\right)$

${lim}_{x\to b^{+}}h\left(x\right)={lim}_{x\to b^{+}}g\left(x\right)=g\left(b\right)$

Given $f\left(b\right)=g\left(b\right)$

$\Rightarrow$LHL $=$RHL

But for $h\left(x\right)$ to be continuous

LHL$=$RHL $=h\left(b\right)$

Since $h\left(b\right)$ value is not known it may or may not be continuous.