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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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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 $$f:Dom(f)\to Range(f)$$

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I want to denote variable, not function. Is variable the same thing as function? –  Sunny88 Feb 22 '13 at 16:15
    
In this notation you specify both. What would you need a variable for on it's own? –  Student Feb 23 '13 at 7:22

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.

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