# The function $f(x)=[x]\cos\big[\large\frac{2x-1}{2}\big]$$\pi where [.] denotes the greatest integer function, is discontinuous at \begin{array}{1 1}(a)\;\text{All x}&(b)\;\text{All integer points}\\(c)\;\text{No x}&(d)\;\text{x which is not an integer}\end{array} ## 1 Answer When x is not an integer,both the functions [x] and \cos\big[\large\frac{2x-1}{2}\big]$$\pi$ are continuous.
$\therefore f(x)$ is continuous on all non integral points.
For $x=n\in I$
$\lim\limits_{x\to n^-}f(x)=\lim\limits_{x\to n^-}[x]\cos\big[\large\frac{2x-1}{2}\big]$$\pi \Rightarrow (n-1)\cos\big[\large\frac{2n-1}{2}\big]$$\pi=0$
$\lim\limits_{x\to n^+}f(x)=\lim\limits_{x\to n^+}[x]\cos\big(\large\frac{2x-1}{2}\big)$$\pi \Rightarrow n\cos\big(\large\frac{2n-1}{2}\big)$$\pi=0$