Question

The function $f\left(x\right)=\frac{\sin \left(\pi \left[x-\pi \right]\right)}{4+{\left[x\right]}^2}$, 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

Answer 1

(a) continuous as well as differentiable for all x ∈ R

Here, $f\left(x\right)=\frac{\sin \left(\pi \left[x-\pi \right]\right)}{4+{\left[x\right]}^2}$

Since, we know that $\mathrm{\pi }\left[\left(x-\mathrm{\pi }\right)\right]=n\mathrm{\pi }$ and $\sin n\mathrm{\pi }=0$.

∵$4+{\left[x\right]}^2\ne 0$

∴f(x) = 0 for all x

Thus, f(x) is a constant function and it is continuous and differentiable everywhere.