Equivalence of Definitions of Derivative

Theorem
Let $I$ be an open real interval.

Let $f: I \to \R$ be a real function defined on $I$.

Let $\xi \in I$ be a point in $I$.

The following definitions for the abelian group are equivalent:

The two forms of the definition of a derivative of $f$ at a point $\left({\xi, f \left({\xi}\right)}\right)$ are equivalent: