# 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

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations: Exercise $4$