Question

Discuss the continuity of the function defined by $f\left(x\right)=|x-5|$.

Answer 1

The given function is $f\left(x\right)=|x-5|=\{\begin{array}{c}5-x,\text{if}x<5\\ x-5,\text{if}x\ge 5\end{array}$

The function is defined at all points of the real line.

Let $c$ be a point on a real line. Then, $c<5$ or $c=5$ or $c>5$

Case I : $c<5$

Then, $f\left(c\right)=5-c$

$\underset{x\to c}{lim}f\left(x\right)=\underset{x\to c}{lim}(5-x)=5-c$

$\therefore \underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$

Therefore, $f$ is continuous at all real numbers less than $5$.

Case II : $c=5$

Then $f\left(c\right)=f\left(5\right)=(5-5)=0$

Then, $\underset{x\to 5}{lim}f\left(x\right)=\underset{x\to 5}{lim}(5-x)=(5-5)=0$

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

$\therefore \underset{x\to 5}{lim}f\left(x\right)=\underset{x\to 5}{lim}f\left(x\right)=f\left(c\right)$

Therefore, $f$ is continuous at $x=5$

Case III : $c>5$

Then, $f\left(c\right)=f\left(5\right)=c-5$

$\underset{x\to c}{lim}f\left(x\right)=\underset{x\to c}{lim}(x-5)=c-5$

$\therefore \underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$

therefore, $f$ is continuous at all real numbers greater than $5$.

Hence, $f$ is continuous at every real number and therefore, it is a continuous function.