The number of points where $f (x) = [sin x + cos x]$ (where []denotes greatest integer function), $x$ $ϵ (0, 2 π)$ is not continuous is :

3

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4

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5

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6

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Solution

The correct option is C $5$
$sin x + cos x = \sqrt{2} (cos \frac{π}{4} sin x + sin \frac{π}{4} cos x) = \sqrt{2} sin (x + \frac{π}{4})$
Thus, $f (x) = [\sqrt{2} sin (x + \frac{π}{4})]$
We know, $[x]$ is not continuous where $x$ is integer
Hence f will not be continuous where $\sqrt{2} sin (x + \frac{π}{4})$ will be an integer
Hence points of discontinuities in the given interval are,
$x = \frac{π}{2}, \frac{3 π}{4}, π, \frac{3 π}{2}, \frac{7 π}{4}$

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The number of points where f(x)=[sinx+cosx] (where []denotes greatest integer function), x ϵ(0,2π) is not continuous is :

The number of points where $f (x) = [sin x + cos x]$ (where []denotes greatest integer function), $x$ $ϵ (0, 2 π)$ is not continuous is :