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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 :-)