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Could you please explain, why the article isn't required before the word function in the following sentence (from a book):

There are two ways to do this. 1. Extend the definition of function.

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It appears that what the author is asking you to extend is not the definition of any particular function, a determinate specific instance, but the definition of the word function. (In linguistic studies we would mark the word with italics, as I have done here, to signal that we are 'mentioning' the word rather than 'using' it to signify something else.)

In effect, the word is the 'name' of the concept, so like a personal name it is already determined and requires no further determiner.

However, it may be (you supply no context) that function may occur here in some such phrase as "function M" or "function qj". In such cases function acts much like a title in a proper name (Mr. Roberts), and is in effect a part of the name—so once more there is no need for a determiner.

You will find the same 'predetermined' construction with most phrases identifying an entity which is, in context, unique: "Room 207", for instance, or "Plate IV", or "English 101".

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    Thank you very much for the reply! Could you clarify, why we need an article before the word function here The value of the function f at a point x is denoted f(x)? – Konstantin Jun 1 '17 at 15:23
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    @Konstantin This is where it gets tricky :) ... You don't need an article; an expression like "function X" may represent either a name or a combination of a common noun and a name - "the function named X". In this example, the author isn't talking about a particular function but about an abstract hypothtetical function; she means something like "The value of the function--let's assume its name is f--at some point --let's designate that point by x--is denoted f(x). – StoneyB Jun 1 '17 at 16:37

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