# How to formally denote that something is a variable

For example, let's say I want to say that in notation $\frac{\partial f}{\partial x}$, $x$ is a variable in real domain. Is there some way to denote it mathematically? For example if I was to say that $x$ is a real number, I would write $x\in\mathbb{R}$. Is there some similar way to denote that $x$ is a variable in real domain?

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In algebra, sometimes you say "Let $R$ be a ring and $R[t]$ with $t$ begin an indeterminant or some variant thereof. Especially in the context when you're dealing with $\Bbb{Q}[x]$ and $\Bbb{Q}[\sqrt{2}]$ near one another. – walkar Apr 29 at 1:43

Yes, if you are looking at functions, you may specify their domain by the notation by$$f:\Bbb R\to \Bbb C$$This notations means that the domain is the reals, and the range is the complex numbers. The general format is $$f:Dom(f)\to Range(f)$$
I don't think you want to do that. The expression $$\frac{\partial f}{\partial x}$$ is just short-hand for a function $g$ having some relation to $f$. Another example is indefinite integrals. Suppose $h$ is an integrable function, then $$\int h(x)\,\mathrm dx$$ just denotes the function (up to a constant) which has a derivative equal to $h$. Here you don't write $x\in\mathbb{R}$ or anything.
If you want to specify the domain of the partial derivative, you could write $$\frac{\partial f}{\partial x}:A\to\mathbb{R},$$ where $A$ is an approriate set.