Question

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

$f(x) = \begin{cases} \frac {x}{|x|}, & \quad \text{if } x \text{<0}\\\ -1, & \quad \text{if } x {\geq 0} \end{cases}$

Answer 1

$f(x) = \begin{cases} \frac {x}{|x|}, & \quad \text{if } x \text{<0}\\\ -1, & \quad \text{if } x {\geq 0} \end{cases}$

It is known that, x<0$\implies$|x| = -x
Therefore, the given function can be rewritten as
$f(x) = \begin{cases} \frac {x}{|x|}=\frac {x}{-x}=-1, & \quad \text{if } x \text{<0}\\\ -1, & \quad \text{if } x {\geq 0} \end{cases}$
$\implies$f(x) = -1 for all x∈R
Let c be a point on the real number.Then, $\lim\limits_{x \to c}$ f(x) = $\lim\limits_{x \to c}$ (-1) = -1
Also, f(c) = -1 = $\lim\limits_{x \to c}$ f(x)
Therefore ,the given function is a continuous function.

Hence, the given function has no point of discontinuity.