If $f: R --- R is a function defined by


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

[.] denotes the greatest

integer function, then $\boldsymbol{f}$ is :

  1. (1) discontinuous only at $x=1$

  2. (2) discontinuous at all integral values of $x$ except at $x=1$

  3. (3) continuous only at $x=1$

  4. (4) continuous for every real $x$

Correct Option: , 4


Doubtful points are $\mathrm{x}=\mathrm{n}, \mathrm{n} \in \mathrm{I}$

L.H.L $=\lim _{x \rightarrow n^{-}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-2) \cos \left(\frac{2 n-1}{2}\right) \pi=0$

R.H.L $=\lim _{x \rightarrow n^{\prime}}[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi=(n-1) \cos \left(\frac{2 n-1}{2}\right) \pi=0$


Hence continuous.

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