Question

Let $f\left(x\right)=x\left[\frac{x}{2}\right]$, for $-10<x<10$, where $[]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

Answer 1

Finding the number of points of discontinuity of $f$

Given, $f\left(x\right)=x\left[\frac{x}{2}\right]-10<x<10$

Here, $\left[\right]$ denotes greatest integer function.

Now $\frac{x}{2}\in \left(-5,5\right)$

We will first check the discontinuity at $x=0$, hence

$$\begin{gathered} f\left(0\right)=0 \\ f\left(0^{+}\right)=0 \\ f\left(0^{+}\right)=0 \end{gathered}$$

Therefore, the function is continuous at $x=0$

Now let us suppose for $x=4$, to check continuity at this point we have,

$f\left(4\right)\ne f\left(4^{+}\right)\ne f\left(4^{-}\right)$

Therefore $f\left(x\right)$ is discontinuous at $x=4$

The same is the case for other values of $x$

Therefore, the function will be discontinuous when $\frac{x}{2}=\pm 4,\pm 3,\pm 2,\pm 1$

Hence the total number of points at which the function $f\left(x\right)$ is discontinuous are $8$