Suppose we have two functions $f, g: \mathbb{D} \rightarrow \mathbb{R}$ where $\mathbb{D} \subseteq \mathbb{R}$. Also, $\forall x \in \mathbb{D}: f(x) = g(x)$.
My question is: Is it possible for such two functions (i.e. can such a pair of functions exist) to have their definitions (i.e. the expressions that describe the relationship between $x$ and $f(x)$ resp. $g(x)$) not equal?
By "definitions not equal" I mean that one expression cannot be transcribed to the other one.
In other, less mathematical words: can there be two functions that are different but give equal values for every point of their domain?
If something is unclear, please comment and I'll update the question accordingly.
EDIT: To exclude trivial examples, let $\mathbb{D}$ be a nontrivial interval or a union of a finite number of nontrivial intervals.
EDIT 2: To narrow it down further more, let $\mathbb{D}$ be induced by the definitions of the functions only, i.e. it covers as much of $\mathbb{R}$ as possible that the functions are still defined. E.g. if $f(x) = \frac{1}{x}$ and $g(x) = \frac{2}{x}$ then $\mathbb{D} = \mathbb{R} \setminus \{0\}$ (I know, in this case they are not equal in values, but it's just for the illustration of what I want from the domain).
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