Differentiability

A function
\[f(x)\]
  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.

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