When I read a math textbook, I encountered:
The condition that a function be continuous is...
I don't know why I need to use "be" instead of "is".
Is it inappropriate if I use "is"? Is there any other word other than "condition" that I need to use "be"?
We could use is to communicate essentially the same idea:
The condition that a function is continuous is (some declaration).
The reason you question the use of be in the fragment is likely that it is an example of a use of the subjunctive mood, which is an often invisible feature of English.
We use the subjunctive to talk about things that are more like mental ideas than factual realities. Dividing utterences into these two categories is often a tricky enterprise.
That a function is continuous is put forth as more of an idea than a factual reality. The idea is issued as a supposition, and we can use the subjunctive mood to communicate such an idea, though we do not always need to.
The subjunctive is often in operation but not noticeable, because in English the subjunctive often appears identical in form to the indicative mood, which (perhaps we can say) more plainly or ordinarily refers to "real", more tangible things.
I will borrow description and examples from the current Wikipedia article on the English sunjunctive, located at:
https://en.m.wikipedia.org/wiki/English_subjunctive
Consider:
He sees a doctor once a year.
This is an example of a use of the verb see in the indicative mood. As you likely expect, see is inflected to sees because it's used with the third-person singular, he.
Compare to:
It is important (that) he see a doctor immediately!
The idea here is different because it describes an opinion or idea that exists in the speaker's mind, not an action or event that we can find in the real world.
Because the subjunctive is in use, we do not change see to sees.
In
It's important that they see a doctor immediately!
We might say that the subjunctive is likely in use because it is likely an opinion or belief, but the form is the same as if it were a use of see in the indicative. In this rather common situation, the operation of the subjunctive is not visible.
Since verbs in the subjunctive appears in unexpected forms only in certain situations, it can surprise and puzzle learners.
One of the ways that the subjunctive appears as noticeable in form is with use of the verb be.
In the example you encountered, be is not changed to is because the grammar of the subjunctive mood differs from that of the indicative.
Yes. There are other verbs with which we use be in the subjunctive and where be does not change form as in the indicative:
I suggest (that) he be taught some manners.
I demand (that) she be promoted.
If that be the case, we should leave.
This may not exactly answer your question specifically on why be is used, but specific jargon may sometimes not be perfect English, consider all the posts on headlinese or song lyrics but they will be perfectly understandable given the surrounding context. Your question is about English grammar when applied to maths, so it's a very left-brain, right-brain combination and you may want to consider that when ruminating about the answers being given.
Here is a link to the maths syntax precedent, maybe it might help in your understanding.
The simple reason be is used instead of is is because there is a
which uses {let, be} as part of the syntax see Section 11.2 and not is.
The condition that a function is continuous is....
The condition that a function be continuous is....
Both the sentences are grammatical, with a slight difference in meaning.
The former expresses the condition that, in fact, makes a function continuous. It's a declaration of facts. In grammar, you say that the verb "is" in the indicative mood.
The latter expresses the condition in a hyothetical or suppositional manner, which may make a function continuous. Grammatically speaking the verb "be" is in the subjunctive mood.
