Is the definite article correct/necessary in sentences like the following?

A line that intersects (the) circle C.
We can see that (the) function f has a maximum at x=0.
Draw a line past (the) edge E. 

I am aware that definite articles must not be used in things like "in Figure 3" or "see Equation 7", because here "Figure" and "Equation" are treated as proper nouns (and thus capitalized). But in other kind of sentences as the previous ones, I am unsure about whether it is also correct to omit the definite article or not.

  • Yes, an article is required in your second example (the zero article in ungrammatical there). The most common and idiomatic choice is the definite article; one conceptual justification for that choice, if you want it, is there is only one function with the definition f. Any other function defined as f is defined would be the same function as f. A rose by any other name would smell as sweet, and so forth. Now, it's possible to use the indefinite article instead, but it's less common to do so. – Dan Bron May 3 '15 at 11:50
  • The last example, with edge E, is a case where you could use the zero article, because E, there, is acting like a proper noun (a named thing, qualified "edge"). The definite article is also fine. – Dan Bron May 3 '15 at 11:53
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    It's more common to say "the function f", but it's possible to just say "function f" with no article. See Ngram. I don't perceive any difference between "edge E" (with no article) and "function f" (with no article). In both cases, E and f are acting like proper nouns (like President Roosevelt). I'd suggest using "the function f", but be consistent. – Peter Shor May 3 '15 at 11:53
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    The above is mathtextbookspeak. In mathtextbookspeak the use of "the" is optional (and usually omitted) in all the cases you list. Nothing wrong as shown. (Consider that "circle C", eg, is essentially a proper noun.) – Hot Licks May 3 '15 at 12:08
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    I agree with @PeterShor and not with Dan Bron: the is optional in the second sentence. – Drew May 4 '15 at 2:11

No, an article is not necessary if they can be proper nouns.

'circle C' is a named circle, and perhaps should even be capitalized as such when there is no article, like 'Miss Jones'. Calling it 'the circle C' may actually be less clear if there are other named circles that may show up later like 'circle D'. After all, 'the miss, Jones' would sound like there are no others in the context with the title 'Miss'.
f(f) or "a function of f" or "the function of f" can be named 'f', but I think it is clearer to name it 'Function f'. In this case, one could make the same argument about capitalizing the word "function". However, in science, terms are case-sensitive and I would never alter the case of the term 'f' for grammatical reasons. 'y=xY' could be different than 'y=xy'. This makes me even more inclined to instead capitalize it as 'Function f', clearly showing it is part of the proper noun. In a sentence I could choose to be verbose and say: "the function of f Function f is shown here", but this would demonstrate that is is also desirable to differentiate these terms with extra text-formatting that you might not use in a different setting (such as color-coding, italics, or bolding). This would also be the case if the sentence was written "the function of f f is shown here"

You have the option to ignore this opinion and only treat 'C' as your proper noun. In that case I would use an article. Also in that case I think it begs for punctuation that was not in your example like: "We can see that the function, f, has a maximum". I would much prefer to see it like: "We can see that Function f has a maximum".
Either way, be consistent throughout. Your intent or perception of which components are actually proper should be reflected in the capitalization, punctuation, and formatting; I attempted to demonstrate this.

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