# Find all points of discontinuity of f, where f is defined by

Question:

Find all points of discontinuity of f, where f is defined by

$f(x)=\left\{\begin{array}{l}\frac{x}{|x|}, \text { if } x<0 \\ -1, \text { if } x \geq 0\end{array}\right.$

Solution:

The given function $f$ is $f(x)=\left\{\begin{array}{l}\frac{x}{|x|}, \text { if } x<0 \\ -1, \text { if } x \geq 0\end{array}\right.$

It is known that, $x<0 \Rightarrow|x|=-x$

Therefore, the given function can be rewritten as

$f(x)=\left\{\begin{array}{l}\frac{x}{|x|}=\frac{x}{-x}=-1, \text { if } x<0 \\ -1, \text { if } x \geq 0\end{array}\right.$

$\Rightarrow f(x)=-1$ for all $x \in \mathbf{R}$

Let $c$ be any real number. Then, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}(-1)=-1$

Also, $f(c)=-1=\lim _{x \rightarrow c} f(x)$

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity.