If it is just a question of definition but not a question of computation, I have heard when I was a student the following definition:
Let $f$ be a real continuous function, class $C^0$. Let
$$
\Delta f(x,x') = \frac{f(x') - f(x)}{x'-x} \quad\mbox{defined on}\quad {\bf R}^2 -\{x=x'\}
$$
If $\Delta f$ admits a continuous extension on the diagonal $\{x=x'\}$ then it is unique, and $f$ is said to be of class $C^1$. The function $f'x) = \Delta f(x,x)$ is then called the derivative of $f$
Of course this is the standard definition, nothing new under the sun, but the $\epsilon-\delta$ calculus if hidden, and of course, not for long :-)
