To be attained at/on?

Question

I'm writing a math paper, so it would probably be easier to answer this question with some background in math. Assume I have a set of "things" and some function turning these things into numbers. I'm interested in two values: the maximum value of this function and the "thing" the maximum is attained on/at. What's the right choice of preposition here? If there are multiple "things" with the same value, do I say "things the maximum is attained on/at" with the same preposition or change it somehow? Also, is this phrase "the thing the maximum is attained on/at" good enough or maybe I'd rather say "the thing on/at which the maximum is attained"?

Answer 1

The choice of preposition is largely dependent on your said mathematical "things":

• The function f attains its maximum value at the point x=0.
• The function f attains its maximum value on the interval [0,1].
• The function f attains its maximum value in the subset M.

Alternatively, one may write

• x=0 is the point at which the maximum value of the function f is attained.
• [0,1] is the interval on which the maximum value of the function f is attained.
• M is the subset in which the maximum value of the function f is attained.

Though, in mathematical writing, one usually shuns beginning a sentence or phrase with a mathematical symbol. Hence

• The point x=0 is the point at which the maximum value of the function f is attained.

is always preferred in mathematical writing. See How to Write Mathematics by Halmos.

If the last formulation sounds repetitive, go with the first set of phrasings.

Answer 2

To attain at: simply because it's an ending point. A point at which you arrive (or try to arrive). Using on makes it sound like a starting point, rather than a maximum (or minimum, both are extremes).
When comparing all of the results, I'd still go for at: the "thing" at which the highest value was attained was X. There's no reason why the adverb should change here.