Equivalence Classes induced by Derivative Function on Set of Functions

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

Let $\mathcal R \subseteq X \times X$ be the equivalence relation on $X$ defined as:
 * $\mathcal R = \left\{{\left({f, g}\right) \in X \times X: D f = D g}\right\}$

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

Then the equivalence classes of $\mathcal R$ are defined as:
 * $\left[\!\left[{f}\right]\!\right]_{\mathcal R} \left({x}\right) = \left\{{g \in X: \exists c \in \R: \forall x \in \R: g \left({x}\right) = f \left({x}\right) + c}\right\}$

That is, it consists of the set of all real functions $f \in X$ which differ by a real constant.

Proof
Follows directly from Derivative Function on Set of Functions induces Equivalence Relation.