Question

$f$ is a continuous function in $[a,b];$ $g$ is a continuous function in $[b,c]$. A function $h\left(x\right)$ is defined as
$h\left(x\right)=f\left(x\right)$ for $xϵ[a,b)$
$=g\left(x\right)$ for $xϵ(b,c]$.
If $f\left(b\right)=g\left(b\right)$, then

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

Answer 1

Answer: (A), (C)

$h\left(x\right)$ has a removable discontinuity at $x=b$
$h\left(b^{-}\right)=g\left(b^{+}\right)=f\left(b^{-}\right)$

The function $h\left(x\right)$ is not defined at $x=b$. However, since $f\left(b\right)=g\left(b\right)$, ${lim}_{x\to b^{-}}f\left(x\right)={lim}_{x\to b^{+}}g\left(x\right)$.
This means, if $h\left(b\right)$ is defined as $f\left(b\right)=g\left(b\right)$, the function will become continuous at $x=b$. So, it has a removeable discontinuity at this point.
Also, by definition of $h\left(x\right)$, $h\left(b^{-}\right)=g\left(b^{+}\right)=f\left(b^{-}\right)$.