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?