Let f:R R is a function defined by f ( x ) = [ x ] ( 2x - 1 2 ) , where [ x ] denotes the greatest integer function , then f is. . .

Let f:R R is a function defined by f ( x ) = [ x ] ( 2x - 1 2 ) ,…

MCQ

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

A
discontinuous only at $ x=0$
B
discontinuous only at non-zero intergral values of $x$
C
continuous only at $x=0 $
✓
continuous for every real $ x$

✓

Answer

Correct option: D.

continuous for every real $ x$

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

Doubtful points are $x=0, n \in I$

${\rm{LHL}} = \mathop {\lim }\limits_{x \to {\pi ^ - }} [x]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi $

$\Rightarrow(n-1) \cos \left(\frac{2 n-1}{2}\right) \pi$

$[x]$ is the greatest integer function.

${\rm{RHL}} = \mathop {\lim }\limits_{x \to {n^ + }} [x]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi $

$ = n\cos \left( {\frac{{2n - 1}}{2}} \right)\pi $

Now values of the function at $x=0$ is $f(n)=0$

Since $\mathrm{LHL}=\mathrm{RHL}=\mathrm{f}(\mathrm{n})$

$f(x)=[x] \cos \left(\frac{2 x-1}{2}\right)$ is continuous for every real $x$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 5. continuity and differentiation→STD 12 - 5. continuity and differentiation — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions