# Alternative definition of derivative

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?

• This is variously called the strong derivative (no relation to "weak derivative" in functional analysis that I know of), the unstraddled derivative, the strict derivative, and the sharp derivative (and apparently also "derivative in the sense of Whitney", according to John B, although this term is new to me). I gave some comments and many references about it in my answer to “Strong” derivative of a monotone function. – Dave L. Renfro Jan 12 '17 at 17:55