Question

The number of points of discontinuity of $f\left(x\right)=\left\{\frac{x}{5}\right\}+\left[\frac{x}{2}\right],x\mathrm{in}[0,100]$ is/are (where $[.]$denotes greatest integer function and $\{.\}$ denotes fractional part function) .

Answer 1

Step 1: Find $\left[\frac{x}{2}\right]$ for $x=0,1,2,3,4$ and $5$ and $\left\{\frac{x}{5}\right\}$ for $x=5$

Here $f\left(x\right)=\left\{\frac{x}{5}\right\}+\left[\frac{x}{2}\right],x\mathrm{in}[0,100]$

Time period of fractional part of $x,\left\{x\right\}=1$

But in the given question, we have, $\left\{\frac{x}{5}\right\}$

Thus, the time period will be$5.$

To plot the graph for greatest integer function,$\left[\frac{x}{2}\right]$, we need to find for each value of$x.$

For $x=0,\left[\frac{x}{2}\right]=0$

For $x=1,\left[\frac{x}{2}\right]=0$

For $x=2,\left[\frac{x}{2}\right]=1$

For $x=3,\left[\frac{x}{2}\right]=1$

For $x=4,\left[\frac{x}{2}\right]=2$

For $x=5,\left[\frac{x}{2}\right]=2$

Now for fractional part,

If$x=5$, then $\left\{\frac{x}{5}\right\}=1$

Step 2: Find the number of discontinuous points

Thus, we can see from the graph, the discontinuity of the fractional part is at,$5,10,15,20,\dots .,100.$

And the discontinuity of the greatest integer function is at $2,4,6,8,\dots 100.$

Hence, we have a total of $20+50=70$ discontinuous points in the graph.

But for both the functions, the multiples of point $10$ will get continuous.

Hence such points are $10+10=20$.

Therefore, we need to remove these continuous points of both the functions, to get the exact numbers of points of discontinuity.

i.e., $70–20=50.$

Therefore, the number of points of discontinuity of $f\left(x\right)=\left\{\frac{x}{5}\right\}+\left[\frac{x}{2}\right],x\mathrm{in}[0,100]$ are$50.$