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When writing some journal paper in scientific or mathematical areas, the usage of comma is very confused. For example,

The objective function f is differentiable when the value of the node x is given.

In the above, f is called the objective function, and x represents the value of the node.

In many papers, the above sentence are written as the below.

The objective function, f, is differentiable when the value of the node, x, is given.

Actually, I have seen both of them in papers. However, I want to know whether there is a specific rule or not.

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    I believe this is a stylistic issue, not a grammatical one. If any “rule” exists, that might be found in (and therefore confined to) the publishing guide of the specific journal. I’d be inclined to leave out the commas but use italics for the variable names: The objective function f is differentiable when the value of the node x is given. – J.R. Nov 10 '17 at 10:13
  • I don't think there is a specific rule either. The first version, with no comma, could be thought of as "f, which is a variable" and the second version, with the comma, could be thought of as "the variable, which is f in this case". But obviously there is virtually no difference in meaning. – stangdon Nov 10 '17 at 14:50
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In researching this answer I learned that I, too, have been using commas incorrectly in this case (to accentuate the subject). I also learned that differentiable is a word even though my spell-checker doesn't recognize it. :)

Correct methods would be:

The objective function f is differentiable when the value of node x is given.

or:

The objective function "f" is differentiable when the value of the node "x" is given.

or even better, on coding sites like StackOverflow:

The objective function f is differentiable when the value of the node x is given.

.

Here is a rule from Purdue OWL:

  1. Do not use commas to set off essential elements of the sentence, such as clauses beginning with that (relative clauses). That clauses after nouns are always essential. That clauses following a verb expressing mental action are always essential.

Thus, the commas around too in my first sentence, above, are allowable since the word too is not essential to the meaning of the sentence, whereas f and x are essential to that sentence.

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The rule here in English is whether you need the f to specify which objective function you are talking about. If you do, say the purpose of putting f is to distinguish it from some other objective function g, then you should not use commas.

However, if you don't need the f for this reason (there is only one objective function you could possibly be talking about) then you should use commas. One very common situation where this is the case in mathematical writing is where you are using the f to give a name to the function: here it is essentially shorthand for "The objective function, which we will call f, ..."

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