Question

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

• A. discontinuous only at $x=1$
• B. discontinuous at all integral values of $x$ except at $x=1$
• C. continuous only at $x=1$
• D. continuous for every real $x$

Answer 1

The correct option is D continuous for every real $x$

Doubtful points are $x=n,n\in I$
$L.H.L.=\underset{x\to n^{-}}{lim}[x-1]\cos \left(\frac{2x-1}{2}\right)\pi =(n-2)\cos \left(\frac{2n-1}{2}\right)\pi =0R.H.L.=\underset{x\to n^{+}}{lim}[x-1]\cos \left(\frac{2x-1}{2}\right)\pi =(n-1)\cos \left(\frac{2n-1}{2}\right)\pi =0$
Also,
$f\left(n\right)=0$
Hence, $f$ is continuous for all real $x$.