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Is the sentence

Sei f eine auf diesem Intervall Funktion.

grammatically correct? If so, what are the rules that govern such constructions and is this sentence better than the following one?:

Sei f eine Funktion auf diesem Intervall.

Another example, but with a demonstrative adjective rather than an article is the sentence

Jede in einem kompakten Intervall stetige Funktion ist beschränkt.

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  • "beschränkt" – Rudy Velthuis Apr 23 '19 at 20:25
  • Did you really mean the first two sentences as written, or is "stetige" (as used in sentence #3) missing? As written, #1 is wrong and #2 is correct; with "stetige" it would be the other way around. It makes a difference whether the adverbial refers to a modifier or the predicate of the sentence. – Endre Both Apr 23 '19 at 20:32
  • @RudyVelthuis Thank you. I correct the typo. – Schroder Apr 23 '19 at 20:47
  • @EndreBoth I intended them as written, but that is perhaps because I did not know including the adjective stetige would make a difference. Could you elaborate on this? – Schroder Apr 23 '19 at 20:48
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Sei f eine Funktion auf diesem Intervall Funktion.

German grammar allows to split up article and noun to put describing elements into those "brace" elements. This braces idea is what governs German word order. However, auf diesem Intervall doesn't describe eine Funktion. It's a place where that thing resides, not a property of that thing.

This changes if you add e.g. definierte to the adverbial. Now it describes a property of the thing itself:

Sei f eine auf diesem Intervall definierte Funktion.

This is because definiert, a Partizip II is used as an adjective and as such, has a tight affinity to the noun. The part auf diesem Intervall is a mere description of definiert in that example.

Jede in einem kompakten Intervall stetige Funktion ist beschränkt.

See the adjective stetig? Same principle, the part in einem kompakten Intervall describes the adjective, the adjective describes the noun so the both belong into the brace spanned by jede and Funktion.

Jede stetige Funktion in einem kompakten Intervall ist beschränkt.

So what about this one? It means a different thing. Be careful. Now you are telling a general "truth" about continous functions in a compact interval, not about functions that are continous in a compact interval.

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2

The first partial sentence sounds crooked. The original,

Sei f eine Funktion auf diesem Intervall

is correct.

The other (partial) sentence is almost correct: beschränkt

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  • What makes these sentence partial? If by partial you mean a sentence-fragment, then don't they each have a subject and verb? – Schroder Apr 23 '19 at 20:49
  • The first one is the beginning of a sentence. A sentence that starts with "Sei" usually does a little more. Like "Be f a function on this interval ... ". I'd expect a "then ... " part and perhaps a few "and ..." parts, here and there. And "Every function, monotonous in a compact interval, is limited" is generally followed by "to ..." or something like that. Otherwise these sentences do not make a lot of sense. Same in German. – Rudy Velthuis Apr 23 '19 at 20:56
  • So what makes the third sentence correct, while the first is not? I am having trouble understanding why I see some sentence with a prepositional phrase in such a location while seeing others with it after the noun. – Schroder Apr 23 '19 at 21:00
  • It is a bit as if I left out the second part of the sentence "If by partial you mean a sentence-fragment". Has a subject and verb too and still sounds incomplete. – Rudy Velthuis Apr 23 '19 at 21:02
  • The first sentence is incorrect because Funktion does not come at the end. The part that you put before should come after it. I am not one of those people who can exactly explain the grammar, but it is wrong. You don't write "Be f an on this interval function" either, do you? You write "Be f a function on this interval". Same principle. ISTM that in the third sentence, the part that comes before Funktion belongs to "stetige" and adjectives generally come before the noun they refer to. – Rudy Velthuis Apr 23 '19 at 21:04
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what are the rules

Word order in noun phrases in German is:

Article - adjective phrases - noun - arguments/modifications of the noun.

(And all prepositional phrases should fall under arguments/modification.)

In your case:

 article |   noun   | modification
  eine   | Funktion | auf diesem Intervall

Word order within adjective phrases is

modifications - adjective

Therefore you get:

        modification          | adjective
 in einem kompakten Intervall | stetig

If you plug this in in the schema above, you get:

article |      modification (adj)      | adjective | noun
 Jede   | in einem kompakten Intervall |  stetige  | Funktion

Is the sentence "Sei f eine auf diesem Intervall Funktion" grammatically correct?

No, because modifications (that aren't adjecives) belong after the noun they modify.

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