## Why do we need continuous functions for differentiation?

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

$\displaystyle \left( {\frac {d}{dx}}f \left( x \right) \right) \left( x_{{0}} \right) ?=?\lim _{h\rightarrow 0}{\frac {f \left( x_{{0}}+h \right) -f \left( x_{{0}}-h \right) }{2 h}}$

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

Thanks!