Question

Show that $f\left(x\right)=|x-5|$ is continuous at $x=5$.

Answer 1

$f\left(x\right)=|x-5|$

Therefore,

$f\left(x\right)=-(x-5),ifx<5$

$=x-5,ifx\ge 5$

For continuity at $x=5$,

Consider, $LHL=li{m}_{x\to 5^{-}}f\left(x\right)=li{m}_{h\to 0}f(5-h)$

$=li{m}_{h\to 0}[5+(h-5\left)\right]$

$=li{m}_{h\to 0}h=0$

$RHL=li{m}_{x\to 5^{+}}f\left(x\right)=li{m}_{h\to 0}f(5+h)$

$=li{m}_{h\to 0}(5+h-5)=li{m}_{h\to 0}=0$

And, $f\left(5\right)=5-5=0$

Therefore, $LHL=RHL=f\left(5\right)$

So, $f\left(x\right)$ is continuous at $x=5$.