Definition of a function's domain and co-domain with subscript in name

I want to define a function that takes a parameter (lets say a real number) and returns a number (lets say a natural number). However, the function makes use of a 'global environment constant variable', lets say $\delta$. It is a constant that does not change once it is defined, so I don't wish to include it in the main list of arguments. However, I am not sure if the notation makes sense in a function definition:

$f_{\delta}: \mathbb{R} \rightarrow \mathbb{N}$

I am not sure if subscripting the $\delta$ in such a function definition is allowed. I would like to use the function later in the text with something like $f_{\delta}(x)$

That seems perfectly fine to me. All you've done is defined an entire family of function from the reals to the naturals (one for each possible value of $\delta$). If you're really worried, you could write something like:
"Let $f\colon\mathbb{R}\times D\rightarrow \mathbb{N}$ be the function...
When the element $\delta\in D$ is fixed, we will often use the notation $f_\delta(\bullet)$ to be $f(\bullet,\delta)$."
• No problem. I would only use my example of phrasing if you really think rigour is totally necessary or if you plan to see how changing the value of $\delta$ affects $f_\delta$. Otherwise, a reader might get bogged down in unnecessary notation. – Dan Rust Aug 14 '13 at 15:48