Question:
The function $f(x)=\frac{\sin (\pi[x-\pi])}{4+[x]^{2}}$, where [.] denotes the greatest integer function, is
(a) continuous as well as differentiable for all x ∈ R
(b) continuous for all x but not differentiable at some x
(c) differentiable for all x but not continuous at some x.
(d) none of these
Solution:
(a) continuous as well as differentiable for all x ∈ R
Here, $f(x)=\frac{\sin (\pi[x-\pi])}{4+[x]^{2}}$
Since, we know that $\pi[(x-\pi)]=n \pi$ and $\sin n \pi=0$.
$\because 4+[x]^{2} \neq 0$
$\therefore f(x)=0$ for all $x$
Thus, $f(x)$ is a constant function and it is continuous and differentiable everywhere,
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