Derivative Function on Set of Functions induces Equivalence Relation

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

This theorem requires a proof. In particular: Based on the result which I can't find which says that $D f = D g \iff f = g + c$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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