I am thinking "C-infinite" or "infinitely differentiable". The second one is absolutely correct, but too long for colloquial use. I'd prefer the first one if possible. Is it correct or common?

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    $\begingroup$ It is pronounced '$C$ infinity'. $\endgroup$ Aug 9 '19 at 6:45

So long as it is clear what $C^\infty$ means in the context where you're speaking, it's perfectly valid to say "C infinity." Otherwise, you might want to establish what you mean by that, but then it should be fair game.

Granted, in my opinion, to say $f \in C^\infty$ as "$f$ is infinitely differentiable" just rolls a little nicer off the tongue. But this is a personal quirk - so long as it's well-established what you mean, it's just a matter of personal preference.

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    $\begingroup$ This is tangential, but it may be worth noting that in various contexts the adjective 'smooth' is used to mean 'infinitely differentiable', as a less wordy alternative. $\endgroup$
    – qualcuno
    Aug 9 '19 at 6:49
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    $\begingroup$ Don’t forget that people write “Let $f$ be $C^∞$ …”, so they essentially also use $C^∞$ as an adjective, in which case saying “cee-infinity” instead of “smooth” is a bit re… But people still say it. I catch myself regularly saying stuff like “Say you have a cee-infinity function …” It’s weird. $\endgroup$
    – k.stm
    Aug 9 '19 at 6:58

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