Definition – The derivative of f(x).

(!) Ordered pair also links here.
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Definition

Let $f$ be a function defined on the set of real numbers.

First, we define $(a_0) := () := emptyset$
Next, we define an ordered pair as follows: $(a_{i-1}, a_i) = {a_{i-1}, {a_{i-1}, a_i}}$ .
Finally, we define a tuple of $n$ elements to be the following set using a recurrence relation:

$(a_1, a_2, ..., a_n) = {(a_1, a_2, ..., a_{n-1}), {(a_1, a_2, ..., a_{n-1}), a_n},~~~$

Where $epsilon > 0$ is a real number.

Notes

This definition is analogous to the definition of a $epsilon$ -Neighborhood of $x$ , except that we do not include the point $x$ itself. This is indicated by the fact that $|x - y| < 0$ . Since $|x - y| neq 0$ (because we haven’t written $geq$ ), it follows that $y neq x$ .