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Suppose there is a function $f$. I already know that it is continuous. Should I write it as "$f$ is continuous" or "$f(x)$ is continuous"? Since $f$ is the function while $f(x)$ is just a value taken at $x$, I would imagine that it would be correct to say "$f$ is continuous over [...]" as continuity is a property of a function. However, on a worksheet that I have, it says $f(x)$ is continuous over $(-\infty,\infty)$. Is that a correct way to write it? Are both forms acceptable? Which one is more "mathematically correct"?

Interstigation
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  • They are both completely fine. It would even be fine not to mention the domain (unless the question asks for it!) - “continuous”, stated without further context, means to say everywhere continuous. – FShrike Mar 28 '22 at 06:01
  • When I write $P(HTH)$ instead of $P({HTH}),$ I am abusing notation for the sake of brevity and because almost no one writes the latter. On the other hand, there is no reason for me to write "$f(x)$ is continuous", even in my personal scribbles. I do largely agree with 5xum's answer though. – ryang Mar 28 '22 at 06:43

1 Answers1

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It depends on the level of rigor you need, which depends on the context in which one is writing.

Technically, it is better to say that $f$ is continuous. This is because, strictly speaking, $f(x)$ is a real number, not a function, and saying a real number is continuous is nonsensical. In papers, books, textbooks and such, I would say it is sloppy to write that "$f(x)$ is continuous".

Less formally, both can be fine, so long as they get the point across unambiguously. The thing is that even if you write "$f(x)$ is continuous over $A$", there is absolutely no ambiguity in what you wanted to say. So, if you are talking about something or writing up something in a really informal note to a colleague, I don't think it's wrong to be a little sloppy.

I don't know how formal the worksheet you have is, so I can't speak on whether the use is OK there. I know that if I wrote something that went to students, I would aim to use the more technically correct language, but we all make mistakes.

5xum
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