Question

If $f\left(x\right)=\cos \pi (\left|x\right|+\left[x\right])$, then $f\left(x\right)$ is/are (where $[\cdot ]$ denotes greatest integer function)

• A. continuous at $x=\frac{1}{2}$
• B. continuous at $x=0$
• C. Differentiable in $(2,4)$
• D. Differentiable in $(0,1)$

Answer 1

Answer: (A), (D)

The correct options are
A continuous at $x=\frac{1}{2}$
D Differentiable in $(0,1)$

We go about doing this problem considering each option.
(a) $f\left({\frac{1}{2}}^{-}\right)=cos\frac{\pi }{2}=0$
$f\left({\frac{1}{2}}^{+}\right)=cos\frac{\pi }{2}=0.$ Hence option a is correct.
(b) $f\left(0^{+}\right)=cos\left(0\right)=1,f\left(0^{-}\right)=cos\left(-\pi \right)=-1$. Hence option b is incorrect.
(c) $f\left(x\right)$ is discontinuous at $x=3$ and thus not differentiable in the given interval.
(d) In $(0,1),f\left(x\right)=cos\pi x$ which is continuous and differentiable.