If f : R$\to$ R is a function defined by $f\left(x\right)=\left[x\right]\cos \left(\frac{2x-1}{2}\right)\mathrm{\pi },$ where [x] denotes the greatest integer function then f is

If x is not an integer then [x] and $\cos \frac{\left(2\mathrm{x}-1\right)\mathrm{\pi }}{2}$ are continuous

$\therefore \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{continuous} \mathrm{on} \mathrm{all} \mathrm{non} \mathrm{integral} \mathrm{points}$

If x is an integer, x=n

$$\begin{gathered} \underset{\mathrm{x}\to {\mathrm{n}}^{-}}{\lim }\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to {\mathrm{n}}^{-}}{\lim }\left[\mathrm{x}\right] \cos \left(\frac{2\mathrm{x}-1}{2}\right)\mathrm{\pi } \\ =\left(\mathrm{n}-1\right) \cos \frac{\left(2\mathrm{n}-1\right)\mathrm{\pi }}{2}=0 \\ \underset{\mathrm{x}\to {\mathrm{n}}^{+}}{\lim }\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to {\mathrm{n}}^{+}}{\lim }\left[\mathrm{x}\right] \cos \left(2\mathrm{x}-1\right)\frac{\mathrm{\pi }}{2} \\ =\mathrm{n} \cos \left(2\mathrm{n}-1\right) \frac{\mathrm{\pi }}{2}=0 \\ \therefore \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{for} \mathrm{all} \mathrm{integers}. \\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{continuous} \mathrm{every} \mathrm{where} \\ \mathrm{option} \left(\mathrm{A}\right) \mathrm{is} \mathrm{correct}. \end{gathered}$$