Question

Find all points of discontinuity of $f$, where $f$ is defined by
f(x) = $\{\begin{array}{c}\frac{x}{|x|}\text{if}x<0\\ -1\text{if}x\ge 0\end{array}$

Answer 1

The given function is $\text{f}\left(x\right)=\{\begin{array}{c}\frac{x}{|x|}\text{if}x<0\\ -1\text{if}x\ge 0\end{array}$
It is known that $x<0\Rightarrow |x|=-x$
Therefore, the given function can be rewritten as
$f\left(x\right)=\{\begin{array}{c}\frac{x}{|x|}=\frac{x}{-x}=-1\text{if}x<0\\ -1\text{if}x\ge 0\end{array}$
$\Rightarrow f\left(x\right)=-1$ for all $x\in R$
Let $c$ be any real number. Then, $\underset{x\to c}{lim}\text{f}\left(x\right)=\underset{x\to c}{lim}(-1)=-1$
Also, $\text{f}\left(c\right)=-1=\underset{x\to c}{lim}\text{f}\left(x\right)$
Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.