This is a doubt about English mathematical formal language.
I would like to know the best way to express a negative hypothesis, in the formulation or statement of a theorem, proposition, etc., using appropriate words, expressions, etc. This theorem is supposed to be included in a scientific paper.
Next, I show a concrete example.
I have proven a theorem. The theorem states that a certain function does not present a local minimum in a concrete part of its domain. This is the current phrasing:
Consider [...] $f$ as a function of $x$, the rest of parameters being arbitrarily fixed. There does not exist a value $x_{\min} \in \,]\,a, b\,[\,$ in which a (local) minimum of $f$ is achieved, that is, verifying
$$ \exists\, \epsilon > 0 \; \text{ such that } \; f(x_{\min}) \leq f(x) \quad \forall\, x \in \,]\,x_{\min} - \epsilon , x_{\min} + \epsilon\,[ \,\text{.} $$
The omitted part "[...]" stands for the meaning of $f$ (for instance: "Consider the final height $f$ as a function of $x$."). In my case, it makes sense the text "as a function of $x$, the rest of parameters being arbitrarily fixed" because the value of $f$ depends on several variables or parameters.
Is this writing or phrasing acceptable? Can it be improved? How?
EDIT:
I am asking this because, despite having checked out some calculus books written in English, it is difficult to find a negative statement in the thesis of a theorem, a proposition, etc., and I want to be sure that my theorem is well written before sending it to any scientific journal.