Derivative Function on Set of Functions induces Equivalence Relation

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Theorem

Let $X$ be the set of real functions $f: \R \to \R$ which possess continuous derivatives.

Let $\RR \subseteq X \times X$ be the relation on $X$ defined as:

$\RR = \set {\tuple {f, g} \in X \times X: D f = D g}$

where $D f$ denotes the first derivative of $f$.


Then $\RR$ is an equivalence relation.


Proof


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