**Question:**

If $f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$, then $f(x)$ is

(a) continuous and differentiable at *x* = 3

(b) continuous but not differentiable at *x* = 3

(c) differentiable nut not continuous at *x* = 3

(d) neither differentiable nor continuous at *x* = 3

**Solution:**

(d) neither differentiable nor continuous at *x* = 3

We have,

$f(x)=|3-x|+(3+x)$, where $(x)$ denotes the least integer greater than or equal to $x$.

$f(x)=\left\{\begin{array}{lr}3-x+3+3, & 2

Here,

$(\mathrm{LHL}$ at $x=3)=\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{-}}(-x+9)=-3+9=6$

$(\mathrm{RHL}$ at $x=3)=\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{-}}(x+4)=3+4=7$

Since, $(\mathrm{LHL}$ at $x=3) \neq(\mathrm{RHL}$ at $x=3)$

Hence, given function is not continuous at $x=3$

Therefore, the function will also not be differentiable at $x=3$