Question

Discuss the continuity of the function $\left[\cos x\right]$ at $x=\frac{\pi }{2}$, where $[.]$ denotes the greatest integer function.

Answer 1

$f\left(x\right)=\left[\cos x\right]$ when $[.]$ is G.I.F

Checking continuity at $x=\frac{\pi }{2}$

$\underset{x\to {\frac{\pi }{2}}^{-}}{lim}\left[\cos x\right]=\underset{h\to 0}{lim}\left[\cos (\frac{\pi }{2}-h)\right]=\underset{h\to 0}{lim}\left[\sin -h\right]=\underset{h\to 0}{lim}[-\sin h]=-1+0$

$\underset{x\to {\frac{\pi }{2}}^{+}}{lim}\left[\cos x\right]=\underset{h\to 0}{lim}\left[\cos (\frac{\pi }{2}+h)\right]=\underset{h\to 0}{lim}\left[\sin h\right]=\left[0\right]=0$

as $\underset{x\to {\frac{\pi }{2}}^{-}}{lim}f\left(x\right)\ne \underset{x\to {\frac{\pi }{2}}^{+}}{lim}f\left(x\right)$

$\Rightarrow f\left(x\right)$ is discontinuous at $x=\frac{\pi }{2}$