Wikipedia:Reference desk/Archives/Mathematics/2022 June 26

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June 26

English equivalent of German "Stelle" when talking about functions?

In the context of numeral systems, German uses the pair of terms "Stelle" and "Wert" for the pair of notions of where a digit is, and which digit it is. Systems based on those notions are in turn designated "Stellenwertsysteme", and correspondingly "place-value systems" in English.

The same pair of terms is used in the context of functions, to designate values taken by the independent and dependent variables respectively. For instance, in f(x) = x², x would be the "Argument", just as in English, but when evaluated with f(3) = 9, 3 is properly the "Stelle". This has the advantage of being able to use simply "Wert" for the 9, instead of needing to specify which of the variables the value is taken by, as in "argument value" versus "function value".

Does English have an analogous pair of terms here as well? I tried googling "place" alongside contextual keywords like "function", "argument", "value", and the results suggest that's not it.

Note that the above is mainly based on my dimming recollections of how one German math teacher presented this, so some of it may be somewhat off or at least non-standard.

Cheers! :)

- 2A02:560:58DB:A900:3936:931D:B81C:32F3 (talk) 16:24, 26 June 2022 (UTC)

You'll often see "input" and "output" used for this.2406:E003:812:5001:3017:14:9751:8F31 (talk) 17:17, 26 June 2022 (UTC)

While the term "place-value notation" is used in English,^[1][2][3] I prefer the term "position" over "place", as in, "a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit", but that preference is a matter of taste.  --Lambiam 19:20, 26 June 2022 (UTC)

I'd never thought about it, but I think I prefer "place" for this sense, and I think that's based on its carrying a clearer connotation of relative rank, from sports usages like "first place". "Position" does have this too, as in "pole position", but more mildly. *shrug*

- 2A02:560:58DB:A900:3936:931D:B81C:32F3 (talk) 21:28, 26 June 2022 (UTC)

In using Stelle referring to "3" in "f (3)", I think it means the place assumed by x on the real number line when the argument x in f (x) is instantiated by taking x to be 3. On the German Wikipedia, at Funktion (Mathematik) § Begriffsgeschichte, we find the sentence, "Auch hier wird noch nicht zwischen der Funktion $f$ und dem Funktionswert $f(x)$ an der Stelle $x$ unterschieden."  --Lambiam 19:52, 26 June 2022 (UTC)

Continuous auto-embedding of R^n -- must it be open?

To make the question concrete, is there any way to have a continuous injection from R³ into itself in such a way that the image is not an open set? Intuitively I can't imagine what such a map would look like, but I haven't been able to come up with a proof there isn't one.

If there is no such map, I think it follows easily that any continuous injection from R³ into itself must be an open map, but none of the links at open mapping theorem really seems to help. --Trovatore (talk) 16:42, 26 June 2022 (UTC)

Isn't this a consequence of Brouwer's invariance-of-domain theorem^[4]?  --Lambiam 19:05, 26 June 2022 (UTC)

Yes, that looks right, thanks. (With that hint, I found that WP does in fact have an article, at Invariance of domain.) --Trovatore (talk) 06:20, 27 June 2022 (UTC)