Derivative

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Let , and let . I define by

I say that is differentiable at , if is nill whenever are both nill.

If is differentiable at , then I define the derivative of at as , for some nill .

Criticism

This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose that , so and . And let's say that we are in a context where anything with absolute value below 1.0 is nill. Let and . Then . This is greater than 1 when , so we reach the seemingly absurd conclusion that is not differentiable where .

This problem is not present in the standard definition of derivative, where the remainder terms go away in the limit and we are just left with .