Definite article in Maths: "(the) function f"

Solution 1:

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.