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
(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 $x=3$
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