For the greatest integer function $f(x)=[x]$, which of the following statements are correct?

A. It is continuous at all real numbers.

B. It is continuous at $x=2$ and 3.

C. It is continuous and differentiable for $x\in(2,3)$

D. It is continuous and differentiable at $x=4.5$

Choose the correct answer from the options given below:

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

If $e^y(x+1)=1$, then

$f(x) = [x]$, where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$, $[x] = 3$

(B) For $2 \leq x < 3$, $[x] = 2$

(C) Right hand derivative of f at $x = 2$ is not defined

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

For the function $f(x) = |x + 1| + |x - 1| + |x - 2|$ which of the following are TRUE?

A. $f(x)$ is continous on $\mathbb{R}$

B. $f(x)$ is not differentiable at $x = 1$

C. $f(x)$ is differentiable on $\mathbb{R}$

D. $f(x)$ has no local maxima

Choose the correct answer from the options given below:

If $y = \log_3(\log x)$, (Consider $\log_e x = \log x$) then the value of $\frac{dy}{dx}$ is

If $f(x) = \begin{cases}\lambda x+5, & if\ x\le 2\\x+3, & if\ x>2\end{cases}$ is continuous at $x=2$, then, the value of $\lambda$ is