Browse Questions

# Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \left\{ \begin{array} {1 1} \large \frac x { |\;x\;| },& \quad\text{ if $x$ < 0 }\\ -1 ,& \quad \text{if$x$$\geq 0$}\\ \end{array} \right.$

$\begin{array}{1 1} \text{The point of discontinuity for this function is x=0} \\\text{There is no point of discontinuity for this function in its domain} \\ \text{ The point of discontinuity for this function is x=1} \\ \text{ The point of discontinuity for this function is x=-1}\end{array}$

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
At $x=0$
LHL=$\lim\limits_{\large x\to 0^-}\big(\large\frac{x}{|x|}\big)$$=-1 f(0)=-1 RHL=\lim\limits_{\large x\to 0^+}f(x) \;\;\;\;\;\;=\large\frac{x}{|x|}$$=-1$
LHL=f(0)=RHL.
$f$ is continuous at $x=0$
Step 2:
Also at $x=c< 0$
$\lim\limits_{\large x\to c}\big(\large\frac{x}{\mid x\mid}\big)$$=-1=f(c)$
Therefore $\lim\limits_{\large x\to c}f(x)=f(c)\Rightarrow$f is continuous at $x=c<0$
Step 3:
At $x=c>0$
$\lim\limits_{\large x\to c}f(x)=1$
Therefore $\lim\limits_{\large x\to c}f(x)=f(c)\Rightarrow$ f is continuous at $x=c>0$
There is no point of discontinuity for this function in its domain.