# Examine the differentiability of f,where f is defined by $f(x)=\left \{\begin{array}{1 1}x[x], & if\;0\leq x<2\\(x-1)x, & if\;2\leq x<3\end{array}\right.$at $x=2$

Toolbox:
• A function is not differentiable if LHL $\neq$ RHL.
• A function is not differentiable if LHL or RHL does not exist.
Given : $f(x)=\left \{\begin{array}{1 1}x[x], & if\;0\leq x<2\\(x-1)x, & if\;2\leq x<3\end{array}\right.$at $x=2$
Consider the left hand limit $x[x]$ if $0\leq x\leq 2$
$\Rightarrow \lim\limits_{\large h\to 0}\large\frac{(2-h)[2-h]-2[2]}{-h}$
This is not defined.
Hence the function is not differentiable.