My question is, why don't we use the following limit:

\left( {\frac {d}{dx}}f \left( x \right)  \right)  \left( x_{{0}}<br />
 \right) <br />
?=?\lim _{h\rightarrow 0}{\frac {f \left( x_{{0}}+h \right) -f<br />
 \left( x_{{0}}-h \right) }{2 h}}

Then we coud have differential quotient for some non-continuous and otherwise non-differentiable points, too.

Thanks!