If f:RR is a function defined by f(x)=[x](2x-12), where [x] denotes the greatest integer f…

Question

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

Accepted Answer

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

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

$L.H.L$ $$ = \mathop {\lim }\limits_{x \to {n^ - }} \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi $$

$$ = \left( {n - 1} \right)\cos \left( {{{2n - 1} \over 2}} \right)\pi  = 0$$

( As $$\left[ x \right]$$ is the greatest integer function )

$$R.H.L. = \mathop {\lim }\limits_{x \to {n^ + }} \,\left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi  = n\cos \left( {{{2n - 1} \over 2}} \right)\pi  = 0$$

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

Since, $L.H.L.$ $$ = R.H.L. = f\left( n \right)$$

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

Solutions · 1