Consider the following alternative definition of the derivative of a function $f:\mathbb R\to\mathbb R$ at a limit point $x$ of the domain of $f$:
$$f'(x)=\lim_{x_1,x_2\to x}\frac{f(x_2)-f(x_1)}{x_2-x_1},$$
where $\lim_{x_1,x_2\to x}\frac{f(x_2)-f(x_1)}{x_2-x_1}$ is the $a\in\mathbb R$ such that for every $\epsilon>0$ there is a $\delta>0$ such that $|\frac{f(x_2)-f(x_1)}{x_2-x_1}-a|<\epsilon$ whenever $x_1$ and $x_2$ are in the domain of $f$, $x_1\neq x_2$, and $\max\{|x-x_1|,|x-x_2|\}<\delta$.
What would happen to calculus if we replaced the usual definition with this one? Could the theory be developed in more or less the same way? Would all the major theorems still hold?
Let me be clear that I am not asking whether this definition is strictly equivalent to the normal one. (In fact, I'm pretty sure it's not: I think that when under the normal definition $f'(x)$ is defined but $f'$ is discontinuous at $x$, $f'(x)$ is not defined under the alternative definition.) I'm asking something a little more vague: could we do pretty much the same thing with this definition?
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$\begingroup$The notion already exists and goes (usually) by the name of derivative in the sense of Whitney.
It is often applied for example on Cantor sets when simply we don't have sufficient data to define the usual derivative, but still we can discuss how a function is locally Hölder continuous or perhaps a bit more (that's where the derivative enters).
On the other hand, there is an important result of Whitney showing (more or less) that after all we get the same. For details see for example: Whitney extension theorem.
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