Question

The function $f\left(x\right)=\left(\right\{x\left\}\right)\sin \left(\pi \right[x\left]\right)$, where [.] denotes the greatest integer function and {.} is the fractional part function, is discontinuous at

• A. all $x$
• B. all integer points
• C. no $x$
• D. $x$, where $x\in Z$

Answer 1

The correct option is B all integer points

Given : $f\left(x\right)=\left\{x\right\}sin\pi \left[x\right]$

Now, greatest integer func. and

fractional part func are

disconinous at integral pts.

So, f(x) is discontinuous at

integral points

Now, let

$k-1<a<k$ $\left[kϵz^{+}\right]$

then $\left[a\right]=k-1$ $\left\{a\right\}=a-\left[a\right]$

$\Rightarrow {lim}_{x\to a}\left\{x\right\}sin\left(\pi \right[x\left]\right)$

$={lim}_{x\to a}xsin\left(\pi \right[x\left]\right)-\left[x\right]sin\left(\pi \right[x\left]\right)$

LHL

$={lim}_{x\to a^{-}}(a-b)sin\pi (k-1)-sin\left(\pi \right(k-1\left)\right)$

$=0$

RHL

$={lim}_{x\to a^{+}}(a+b)sin\pi (k-1)-(k-1)sin\left(\pi \right(k-1\left)\right)$

$\Rightarrow f\left(x\right)$ is discontinuous at only

integral points.