Students like to categorize notations to pin down their understanding of exactly what these notations stand for. Thus, given the expressions $f(x_{0})=f(x)|_{x\leftarrow x_{0}}$, $x=x_0+h$, or $lim_{x\rightarrow x_{0}}$, they like that the notation $x$ is in the category of (global) variables and that the notation $h$ is in the category of (local) variables. And, while they do understand that the notation $x_0$ stands for a number that is fixed "for the duration", they want to know what is the (name of the) category of "notations" that $x_0$ falls in.
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9$\begingroup$ I encountered this question on the review queue---at least one user has voted to close it. I chose not to vote-to-close, but I understand the impulse. As it is currently written, your question might be an appropriate question for Mathematics, with the "terminology" tag. Could you please edit your post to highlight the educational issue you are trying to highlight? $\endgroup$– Xander Henderson ♦Commented May 21, 2020 at 12:20
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2$\begingroup$ Never heard "local variable" applied to math expressions. $\endgroup$– Rusty CoreCommented May 21, 2020 at 19:11
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2$\begingroup$ @XanderHenderson: Not OP but I can absolutely see some educational issues being related to this query. For instance, the limit definition of a derivative, $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$, always confuses students because we use it to show that, for instance $f'(a)=2a$ when $f(x)=x^2$, so then we turn around and say $f'(x)=2x$. But that's a different $x$ than the one in the limit ... I also wish we had some terminology to use with students regarding these things! $\endgroup$– Brendan W. SullivanCommented May 21, 2020 at 19:55
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3$\begingroup$ @BrendanW.Sullivan It is not that I don't see an education / pedagogy question somewhere in what is written, only that as the question is currently written, the educational question is non-obvious, and it is not clear whether the asker is concerned with how a term might be used in the classroom, or how it might be used in research. As I said, I didn't vote-to-close, but some clarification would be good. $\endgroup$– Xander Henderson ♦Commented May 21, 2020 at 20:56
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1$\begingroup$ @BrendanW.Sullivan I agree is well. I wonder if $h$ should be thought of as a parameter in this context, fixed to well defined limit where $h=a$. For example, we often define functions that depend on notionally suppressed parameters like $a$, for example $g(x) = \frac{f(x) - f(a)}{x-a}$. We may then specialize these function that certain values like $a=1$ or $a=4$. A limit is then related to a specialization of the parameter to a particular value, say $0$ (infinite limits are a bit trickier). I feel like this squares with the $\delta-\epsilon$ definition, where $\delta$ would be a parameter. $\endgroup$– Nate BadeCommented May 24, 2020 at 23:41
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In this instance I would probably say "Fix an (arbitrary) point $x_0$" and carry on. The idea is it's initially unknown but after its introduction stays constant which this sounds like it conveys pretty nicely, and I don't think it's worth splitting philosophical hairs over what is a "variable" vs. "parameter", especially in the face of how scope and discharging of variables are rarely precisely dealt with in typical mathematical writing.
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$\begingroup$ That is an explanation of what $x_0$ stands for. I hope the edit makes now clear what it was I was after. $\endgroup$ Commented May 24, 2020 at 20:21