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.