QJEEM21Mad757: JEE Main 2021 Mathematics - Class 12 JEEM21Mad757 Solution: If f : R R is a function defined by f(x)= [x - 1] , where [.]…

Question

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If f : R

\to

R is a function defined by f(x)= [x - 1]

\cos \left( {{{2x - 1} \over 2}} \right)\pi

, where [.] denotes the greatest integer function, then f is :

Accepted Answer

Given,

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

where [ . ] is greatest integer function and f : R

\to

R

\because

It is a greatest integer function then we need to check its continuity at x

\in

I except these it is continuous.

Let, x = n where n

\in

I

Then

LHL =

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

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

RHL =

\mathop {\lim }\limits_{x \to {n^ + }} [x - 1]\cos \left( {{{2x - 1} \over 2}} \right)\pi

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

and f(n) = 0

Here,

\mathop {\lim }\limits_{x \to {n^ - }} f(x) = \mathop {\lim }\limits_{x \to {n^ + }} f(x) = f(n)

\therefore

It is continuous at every integers.

Therefore, the given function is continuous for all real x.

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