Question

F(x) = \[[\sin x]\], Where \[[\text{  } ]\] denotes the greatest integer function, is continuous at.

a) $\dfrac{\pi }{2}$ 

b) $\pi $

c) $\dfrac{{3\pi }}{2}$ 

d) $2\pi $

Answer 1

Hint: Find the points where the given function that is the greatest integer function is discontinuous and try to draw the graph of \[[\sin x]\] and find the discontinuous points. Take intervals of $\dfrac{\pi }{2}$ each to check the continuity and also to draw the graph.

Complete step by step answer:

Given function is  F(x) = \[[\sin x]\].

We know that F(x) = [x] is discontinuous at integral values of x.

In the interval \[[0, \dfrac{\pi }{2}]\], sinx is continuous and its range varies from [0,1]

Therefore in the interval \[(0, \dfrac{\pi }{2}), [\sin x]\] is continuous and is equal to 0

At $\dfrac{\pi }{2}$ , $\sin x$ is 1 therefore \[\sin x]\] is equal to 1 and \[[\dfrac{\pi }{2} , \pi ]\] $\sin x$ is continuous and its range varies from [1,0]

Therefore in the interval \[(\dfrac{\pi }{2} , \pi ], [\sin x]\] is continuous and is equal to 0

In the interval \[[\pi ,\dfrac{{3\pi }}{2}], \sin x\] is continuous and its range varies from [-1,0]

Therefore in the interval \[(\pi ,\dfrac{{3\pi }}{2}), [\sin x]\] is continuous and is equal to -1

Therefore we can say that at x=$\pi $, \[\sin x\] is discontinuous.

And at x=$\dfrac{{3\pi }}{2}$,\[\sin x \] is equal to -1.

In the interval \[[\dfrac{{3\pi }}{2},2\pi ]\] , $\sin x$ is continuous and its range is [0,-1].

Therefore in the interval \[(\dfrac{{3\pi }}{2},2\pi ),[ \sin x]\] is continuous and is equal to -1.

Therefore we can say that at \[x=\dfrac{{3\pi }}{2} ,[\sin x]\] is continuous and is equal to -1 and

At x=\[2\pi ,[\sin x]\] is equal to 0 therefore it is discontinuous at $x=2\pi $.

Therefore we can say that at \[x=n\pi  ,\dfrac{(4n + 1)\pi }{2} [\sin x]\] is discontinuous where n is in integers.

The one and only value that satisfies the above condition and is in the given options is $\dfrac{\pi }{2}$.

So the correct option is (a).

Note:
Make sure to check all the intervals and values properly so that you don’t make any mistakes. Check the open interval and closed intervals conditions also so that you can find it whether it is continuous or not in the given interval.