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
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 $4$