Use of Being in a Sentences

Solution 1:

First, let me address the issue of whether the sentence is correct (i.e. acceptable, i.e. 'natural').

A simplified version of the construction you are asking about is this:

[1] ^?This is a proof for the function f being continuous.

I'm putting a ^? in front because to me, [1] does not quite sound natural. To test this intuition, I searched google books for "proof for * being" (or "implication for * being"). There were only a few hits, but they were either translations (here and here) or works by speakers of Sri Lankan English (here), which has somewhat separate rules from Standard English. On the other hand, when one searches for "proof for," one finds that the complement of for is (except for the several hits I just mentioned) invariably a noun phrase (NP), e.g. a proof for determinism, a new proof for a result of Kingan and Lemos, a proposed proof for a given statement, etc. Thus, instead of [1], it is more idiomatic to have

[2] This is a proof for the continuity of function f.
or
[3] This is a proof that the function f is continuous.

Having said all that, if proof is replaced by evidence, the construction is unquestionably idiomatic:

[4] This is evidence for function f being continuous.

I don't think it is possible to find some principled reason why [1] is dubious but [4] is OK. That's how languages work. The best I can do is provide evidence that the construction in [4] is indeed fully idiomatic. This is easy, because searching google books for e.g. "evidence for it being" and "is evidence for * being" returns many hits. Here are just a few of them:

One piece of evidence for it being a dream was that his memories seemed to be a movie. (source)
The evidence for it being an aetiological factor is weak. (source)
There is ample evidence for it being both a concept and a percept at once. (source)
Someone's being psychologically continuous with you is strong evidence for her being you. (source)
This in turn might reasonably be taken as evidence for Leibniz being a kind of A-theorist. (source)

Now that we have found acceptable sentences which feature the for + being construction, we can turn to analyzing the grammar of it.

Let's consider sentence [4]. According to the terminology used in CGEL, evidence for function f being continuous is a noun phrase (NP). In that sentence, its grammatical function is that of a predicative complement. But it could have other functions, such as

[5]  i.  There is evidence for function f being continuous. (displaced subject in
                                                                                                           an existential clause)
      ii.  I require evidence for function f being continuous. (direct object)
     iii.  He proceeded without evidence for function f being continuous. (complement
                                                                                                                                  of a preposition)

and so on.

Now let's analyze the NP

[6] evidence for function f being continuous.

It consists of a head, which is the noun evidence, and of the preposition phrase (PP) for function f being continuous. The PP is an internal dependent of the NP. In this case it is a complement rather than an adjunct, because it expresses a semantic argument of the head noun (CGEL, pp. 441-443): it is what evidence helps to establish.

Let us now look at the structure of the PP

[7] for function f being continuous.

It consists of the head, which is the preposition for, and its complement, which is the gerund-participial clause

[8] function f being continuous.

The clause [8] is non-finite, and its head is a non-finite form of the verb to be, being. Its finite counterpart is function f is continuous. If the subject were a pronoun, it would likely be in the genitive case: its being continuous (however, it being continuous is also acceptable). If the pronoun were she or he, in this case it is pretty much obligatory to use the genitive: This is evidence for her/him being continuous.