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

Option: 1

continuous for every real x


Option: 2

discontinuous only at x = 0


Option: 3

discontinuous only at non-zero integral values of x


Option: 4

continuous only at x = 0


Answers (1)

best_answer

f:R\rightarrow R,f(x)=\left [ x \right ]cos\left ( \frac{2x-1}{2} \right )\pi

=\left [ x \right ]cos\left ( \pi x-\frac{\pi }{2} \right )=\left [ x \right ]sin\pi x

Let\; n\; be \; an\; integer.

\lim_{x\rightarrow n^{+}}f(x)=0,\lim_{x\rightarrow n^{-}}f(x)=0

\therefore \; \; f(n)=0

\Rightarrow \; f(x)\; is\; continuous\; for\; every \; real\; x

Posted by

Kuldeep Maurya

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