Alternatively one can define differentiability of \(f\) at \(x_0\), by claiming the existence of a function \(\Delta\) which is continuous in \(x_0\), such that
\begin{equation*} f(x)=f(x_0)+(x-x_0)\Delta(x). \end{equation*}Indeed, \(\Delta(x_0)=f'(x_0)\).
Remark
- Note for every \(x_0\) there is another \(\Delta\). So this does not mean that for a differentiable function the derivative is continuous. This is not true in general.
- The above given representation implies that \(f\) is continuous in \(x_0\).