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Consider a function f(x,y) defined on X x Y.
One says that f(0,y) is the evaluation of f at 0 ...
My problem using searching engines is that usual occurrence of evaluation is evaluation at certain point, but if the function in question has multiple arguments, then which preposition helps to specify the argument it is evaluated in/at?
I am not sure that there is a unique way to translate f(0, y) into English. The whole point of mathematical notation is to provide a succinct and less ambiguous form of communication than any natural language. Moreover, there are usually many ways to convey the same thought in a natural language.
In most cases, I would avoid the word "evaluation" because it implies a numeric answer. If there is no numeric answer, "evaluation" creates a misleading expectation. "f(0, y) is the simplification of f(x, y) in the special case that x equals zero" seems to me to work, but there are probably hundreds of ways to express the same thought in English.
Math has its own unique way of saying things, so this isn't really a general-purpose English question. I'm not sure if "evaluate" is the right verb to use when talking about defining a function of two variables when only one variable is known, as the result is another function, not a specific value.
Which is why I have some trouble figuring out what to say, since it's not clear what you're trying to do. For example if I were to try to explain the problem I might say something like:
Given a function f(x,y), then f(0,y) is a reduction of the function when x=0. Similarly we can say f(n,y) reduces/simplifies the function when the first argument is known to be n.
My verbiage is probably a little strange, as it's been a long time since I've taken any serious math courses. Someone who does this on a regular basis could better tell you what terms would be common.
I assume from the context that this question concerns a function from two-dimensional real space (RxR) into the real line R. That assumption helps me explain my answer but does not change it.
According to Burkill and Burkill A Second Course in Mathematical Analysis, the word that describes a function, such as f(0, y), which takes values that are identical to another function (in this case f(x, y)) but on a subset of the latter's domain (in this case the line x=0 is a proper subset of RxR) is the restriction of f to the the subset.
I daresay there are other terms used by other authors. In quite a few years of studying mathematics I myself have not encountered in this context either 'simplification' (pace @JeffMorrow) or 'reduction' (pace @Andrew) but it is quite possible that they are used also.
If you want to refer specifically to the values taken by f(0, y), they are its range.
