Notation for a partially evaluated $n$ dimensional function I can’t seem to find the right phrasing to google this question. Suppose I have function $$f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$$
you might denote the evaluation of $f$ at point $(x,y)$ as $f(x,y)$. For any $x_0\in\mathbb{R}$, you can make a function $g:\mathbb{R}\to\mathbb{R}, y\mapsto f(x_0,y)$.
Is there any standard notation for this function $g$ (which refers to $f$ and $x_0$ in some way)?
(Note: I want to use this notation (above) to turn a function with $n$ variables mapping into a ($n$-$m$) dimensional codomain, the function $f$ above is just an example)
 A: As mentioned in the comments, it is valid to use the notation $g= f(x_0,\cdot)$. However, you are correct that this isn't generally practical when we are dealing with many variables. Another (in my opinion better) way to denote a function like $g$ is as follows.
Let's consider the example some function $f: \mathbb{R^3} \rightarrow \Omega$ where $\Omega$ could be anything, it doesn't really matter. Then we can define the functions $f_{x_0} := f(x_0, y, z)$ and $f_{y_0} := f(x, y_0, z)$ and $f_{z_0} := f(x, y, z_0)$. This allows you to hold one variable constant much more easily when you are dealing with functions of an arbitrary number of variables (note that $x_0, y_0, z_0$ are all held constant and are not variables in the above functions).
Another example to generalise the above would be to consider a function $h: \mathbb{R}^n \rightarrow \Omega$ and define
$$h_{x_{i_0}} := h(x_1, ... x_{i_0} . . . x_n)$$
This allows you to take any function with a finite dimensional domain and define a new function from this where we hold one of the variables in the domain constant (as before, $x_{i_0}$ is held constant in the above example and so it is not a variable).
As mentioned in the comments, there is some ambiguity regarding this notation when it is being evaluated at a specific value. The common workaround (that I am not personally a fan of) is when dealing with a function of two variables to use the alternative notation of $f_{x_0}$ and $f^{y_0}$. However, the reason that I am not a fan of this notation is that it can be mistaken for being an exponent and also does not generalise to a function of more variables very well.
The notation that I have always preferred is to still denote the function as $h_{x_{i_0}}(x_1 . . . x_n)$ in this way, however, when we are evaluating at a specific value of $x_{i_0}$, we use the less common (but much clearer) notation of $h_{[x_{i_0} = \space n]}$ for some fixed value $n$.
Alternatives
For completeness, here are some other ways to address your concern.


*

*We can use evaluation notation. Where we define a new function $g$ such that:
$$g_{(i,\alpha )} := f(x_1 \space . . . \space  x_{(i-1)}, \space \alpha , \space x_{(i+1)} . . . x_n )  $$
Here the i$^{th}$ variable is replaced with the constant $\alpha$.





*You can use the "restriction" notation $f\space |_{(x_i = 5)}$. This can become quite cumbersome and so you can define this as a new function. However, if you do this then it transposes into an equivalent definition of the function $g$ in ($1$).


Ultimately, I feel as though there are shortcomings to both of these approaches as it becomes notationally difficult to define $n$ different functions that are each responsible for fixing one of the $n$ variables in the domain.
Therefore, the notation that I would recommend would still be that which I recommended in the original answer. However, these are all valid alternatives and have all been used before in this context.
