3
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Find Number of discontinuities of $$f(x)=\lfloor\sin ^{-1} x\rfloor+\lfloor\cos ^{-1} x\rfloor$$

My try:

The floor function $\lfloor x\rfloor$ is discontinuous when $x \in \mathbb{Z}$

Now the integer values $\sin^{-1}x$ takes are ${-1,0,1}$

This means the function $h(x)=\lfloor \sin^{-1} x\rfloor$ is discontinuous at

$x=-\sin 1 , 0, \sin 1$

Similarly the integer values $\cos^{-1} x$ takes are $0,1,2,3$

This means the function $g(x)=\lfloor \cos^{-1} x\rfloor$ is discontinuos at $x=\cos 3, \cos 2,\cos 1,1$

but since the domain of $f(x)$ is $[-1, \:\: 1]$

we need to check Left continuity of $f(x)$ at $x=1$

$$\lim _{x \to 1^-}f(x)=\lim_{h \to 0} \lfloor \sin^{-1}(1-h)\rfloor+\lfloor \cos^{-1}(1-h)\rfloor=1=f(1)$$

hence $f(x)$ is continuous at $x=1$

hence number of discontinuities are $6$

is this right approach?

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  • 2
    $\begingroup$ There's no indication of why the function shouldn't be continuous at those points. You only found the points where the discontinuity can happen, but you need to check if the function really is discontinuous at those points. $\endgroup$ – Jakobian Dec 5 '18 at 17:34

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