Question

If $f:R\to R$ is a function defined by $f\left(x\right)=[x-1]\cos \left(\frac{2x-1}{2}\right)𝜋$ , 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 $x$

Explanation for the correct options:

Finding the value of $f$:

$f\left(x\right)=[x-1]\cos \left(\frac{2x-1}{2}\right)𝜋$

For $x=n,n\in Z$

Left Hand Limit $=\underset{x\to n^{-}}{\lim }f\left(x\right)$

$$\begin{gathered} =\underset{x\to n^{-}}{\lim }\left[x-1\right]\cos \left(\frac{2x-1}{2}\right)\mathrm{\pi } \\ =0 \end{gathered}$$

Right Hand Limit $=\underset{x\to n^{+}}{\lim }f\left(x\right)$

$$\begin{gathered} =\underset{x\to n^{+}}{\lim }\left[x-1\right]\cos \left(\frac{2x-1}{2}\right)\mathrm{\pi } \\ =0 \end{gathered}$$

and $f\left(n\right)=0$ $\left[\because \cos \left(\right(2k+1\left)\pi \right)=0,k\in I\right]$

$\Rightarrow$LHL $=$ RHL $=f\left(n\right)$

$\therefore f\left(x\right)$ is continuous for every real $x$.

Hence, Option ‘D’ is Correct.