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 $f(\bullet,\delta)$."