\[f(x)\]
is differentiable at a point \[x_0\]
if the first derivative is continuous at that point.\[f(x)= | x |\]
is not differentiable at \[x=0\]
because\[\frac{df(x))}{dx}= \left\{ \begin{array}{c} -1 \; x \lt 0 \\ 1 \; x \gt 0 \end{array} \right. \]
and
\[\frac{d(f(x))}{dx}\]
is not defined at \[x=0\]
. \[f(x)\]
is differentiable on \[\mathbb{R} - \{ 0 \}\]
.\[f(x)= | x^2 |\]
is differentiable everywhere because\[\frac{df(x))}{dx}= \left\{ \begin{array}{c} - | x | \; x \lt 0 \\ | x | \; x \gt 0 \end{array} \right. \]
and
\[\frac{d(f(x))}{dx}=0 \]
at \[x=0\]
so the first derivative is continuous and \[f(x)\]
. is differentiable everywhere.