In traditional grammar the verb BE was considered as a main verb (or lexical verb) when used on its own in a sentence. It was only considered an auxiliary when it was used as part of a passive construction or a continuous construction. However, we now understand that auxiliaries are a grammatical class of words that have the same grammatical properties. Lexical or main verbs do not share these properties. But the verb BE usually does, even when it is the only verb in the sentence.
The important central properties which characterise auxiliary verbs are sometimes referred to as NICE properties. NICE is an acronym for:
Auxiliary verbs are necessary in English to make negatives with not. They also have a tendency to contract with the word not. If there is no auxiliary in the positive version of the sentence we use the dummy auxiliary DO:
- The giraffes can't stand up.
- We don't know what to do.
- They mustn't leave.
- The elephants aren't happy.
In the sentences above we see the negative particle occurring attached to the auxiliaries. When not contracted it must occur directly after the auxiliary in canonical negative sentences. Notice how the verb BE is fulfilling this function even though there is no following verb.
There are several instances in English when we need to invert the subject and auxiliary verb. The most obvious is in yes/no questions. Obviously we can only do this if there is an auxiliary! When there is no normal auxiliary in the declarative counterpart sentence we use the auxiliary DO:
- Can the giraffes stand up?
- Do we know what to do?
- Must they leave?
- Are the elephants happy?
Notice again that the verb BE is behaving just like the other auxiliary verbs.
Sometimes we delete material following an auxiliary verb leaving the auxiliary to stand in for the missing material. In answer to the questions above we can give the following answers. The missing material is in brackets:
- They can! [stand up]
- We do. [know what to do]
- They must! [leave]
- They are. [happy]
Again BE is not an exception here.
When wanting to add positive emphasis to a sentence, underlining that the sentence is true, we add emphasis by stressing the auxiliary, thereby giving it a full vowel (and preventing any contractions with subject pronouns). If there is no auxiliary in the canonical sentence we need to use the auxiliary DO:
- The giraffes can stand up.
- We do know what to do.
- They must leave.
- They are happy.
We can see then that BE is a member of this auxiliary verb family.
The Original Poster's question
The Original Poster is right that the word the is a function word.
Because it is an auxiliary, the verb BE should be regarded as a function word too. The Original Poster is a skilled English pronunciation enthusiast (this question comes from a book on pronunciation). The verb BE behaves like a function word in terms of pronunciation too.
The verb BE, like many other function words usually doesn't take stress, although like other auxiliaries it is always stressed when part of a negative contraction:
- He is 'happy
- He isn't happy
Like other auxiliaries it is often stressed when stranded at the end of a clause - even when it is not negative:
Like other function words, the different grammatical forms of BE usually have two pronunciations. There is a normal one (referred to by linguists as the weak form) for when the word is not stressed and not stranded. There is also a strong form for when these words are either stressed or stranded or both. For example the word am, when not contracted is usually pronounced /əm/. But if it is the last word or if it is stressed, it is pronounced /æm/:
- Whose coming? I am! (/æm/ because it is the last word, even though it is not stressed here)
- I am coming! (/æm/ because stressed in emphatic sentence)
- I alone am able to help you. (/əm/ in normal unstressed occurrence)
It is useful then in terms of English phonetics and phonology to regard BE as a function word.