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.