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What do we call the (proper-class) function $$\mathrm{eval}(*,*)$$ such that for all functions $g$ and all $x \in \mathrm{dom} \;g$ we have $\mathrm{eval}(g,x) = g(x)$ ? I looked up 'evaluation function' but it seems to be a concept in game theory as opposed to the concept I'm looking for.

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So what if there's a clash? "Normal function" has a meaning in set theory and in probability theory. I can assure you, the two notions do not coincide. –  Asaf Karagila Aug 17 '13 at 3:18
@AsafKaragila, I don't care if there's a clash, but I can't get more information on the function until I know its standard name. –  goblin Aug 17 '13 at 3:20
What sort of information would you like to find? –  Asaf Karagila Aug 17 '13 at 3:20
@AsafKaragila, in particular, I'm wondering if we can get away with only one $\mathrm{eval}$ function. Like, can I write $\mathrm{eval}(\mathrm{eval},(f,x))$ ? This sort of thing doesn't work in ZFC as I'm sure you're well aware, but maybe in other systems. –  goblin Aug 17 '13 at 3:23
Alternatively, you could define $$\text{eval}(x,g) = [\text{eval}(x)](g) = g(x)$$ That is, we would write $$\text{eval}=\lambda x.[\lambda f.[f(x)]]$$ –  Omnomnomnom Aug 17 '13 at 3:33

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