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If, say, a function $f\colon(a,b)\to\mathbb R$ is continuous and there's a continuous function $g\colon[a,b)\to\mathbb R$ such that $f$ and $g$ agree on $(a,b)$, then some German textbooks call $f$ "stetig fortsetzbar" in $a$, and $g$ is called the "stetige Fortsetzung". Of course, "stetig" is for continuous, but is there a standard English term for this concept? Is it maybe "continuously continuable"?

[FWIW, the more general definition is that a function $f$ defined on a set $D$ is "stetig fortsetzbar" in $x_0\neq D$ if $\lim_{x\to x_0}f(x)$ (is defined and) exists.]

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    $\begingroup$ 'Continuous extension', I guess. $\endgroup$ Sep 17, 2022 at 8:13

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stetige Fortsetzung: continuous extension

stetig fortsetzbar: continuously extensible

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    $\begingroup$ Oh boy, "extension" somehow didn't come to mind, but a quick web search seems to confirm this. That's interesting because the translation of "analytische Fortsetzung" is "analytic continuation" and not "analytic extension". $\endgroup$
    – Frunobulax
    Sep 17, 2022 at 8:24
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    $\begingroup$ ... you cannot translate too literally ... Fortsetzung has two (or more) English translations. Outside mathematics you can use whatever translation you wish. But in mathematics, we fix certain terminology and always say it the same way. $\endgroup$
    – GEdgar
    Sep 17, 2022 at 10:01
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    $\begingroup$ @GEdgar Correct, the German word Fortsetzung has a range of meanings depending on the context (continuation, sequel, extension, instalment, ...). When used in mathematics it normally means extension. Similarly the word "stetig" can mean constant, steady, continuous, ... The combination "stetige Fortsetzung" is only used in mathematics (at least I have never heard it in another context) and it means continuous extension. See for example en.wikipedia.org/wiki/Tietze_extension_theorem and de.wikipedia.org/wiki/Fortsetzungssatz_von_Tietze. $\endgroup$
    – Paul Frost
    Sep 17, 2022 at 11:42
  • $\begingroup$ @GEdgar Of course I know that one cannot simply translate and that there's a fixed terminology in mathematics. That's why I asked here in the first place. If I had just needed an English translation for the word "Fortsetzung", well, I have a dictionary at home... $\endgroup$
    – Frunobulax
    Sep 18, 2022 at 9:44
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    $\begingroup$ Plus: "continuous continuation" would be strange. $\endgroup$
    – GEdgar
    Sep 18, 2022 at 10:47

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