Proper punctuation after cases environment

I know that when writing a displayed equation, it's correct to write punctuation before the closing \] as follows:

This proves $x^2 = 3,$ and it follows that $x = \pm\sqrt{3}.$


If I'm using a cases environment, where's the correct place for the period to end the sentence?

Absolute value is defined as $\lvert x \rvert = \begin{cases} x & \text{if x \ge 0} \\ -x & \text{if x < 0} \end{cases}$


If I put it after the cases environment, it looks funny. If I leave it off completely, then it also looks bad. It looks okay to put it at the end of the second line ($x < 0$.}), but are there any better solutions?

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You do know that using  in LaTeX is not a good idea, and that you really should stick to $...$, I assume. –  Joseph Wright Jul 28 '10 at 18:28
I've actually heard that before, but why is that? Just asked that here: tex.stackexchange.com/questions/503. –  Ben Alpert Jul 28 '10 at 18:33

You've pretty much enumerated all the sensible options in your question, so I don't know that there's anything else. What I've seen sometimes (IIRC) is to put a comma after each case, and then I guess the period could go at the end of the last one.

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Thanks. I think that's what I'll do. –  Ben Alpert Jul 31 '10 at 20:54

This is what I would do:

Absolute value is defined as:
%
$\left\lvert x \right\rvert \coloneqq \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$


Thus: a colon introducing the maths, and no punctuation afterwards. My reasoning is in my answer to ... err ... I thought we had a question on punctuation after maths here? What happened to it? Nonetheless, it was asked over on MathOverflow so I can cite it as this answer instead.

My basic point is that anyone reading more than a few symbols will need to stop "reading English" and switch to "reading maths" and that this mental switch has the same effect as a full stop. Of course, you can punctuate internally to the maths since punctuation still has meaning, but it should be read as being internal to the mathematics and not refer to the external text. Thus David's suggestion in his last sentence (at time of writing) makes sense to me.

Those that find the absence of a full stop abhorrent (there are some, as can be seen by looking at the comments to my answer on MO) can adjust the preceding remark to make it a full sentence, or to make it clearer that it isn't. Thus:

Absolute value is defined as in the following formula.
%
$\left\lvert x \right\rvert \coloneqq \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$


(You may notice that I've also introduced a couple of other stylistic "improvements"!)

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One of your stylistic improvements I don't see the rationale for: why the %? –  Mark Meckes Jul 29 '10 at 13:55
@Mark Meckes: not all my improvements are aimed at the end product. In that case, putting in the % clearly separates the mathematics from the running text, making it easier to spot in the source file. Thus the rationale is to make life easier on yourself. A good rule of thumb is to make life easier on the you reading the source file in 12 months' time when you get the referee's report after having forgotten all about that paper in the meantime. –  Andrew Stacey Jul 29 '10 at 14:17
Oh, I understand all that. I just would have thought that putting the \[ on a line by itself would achieve that end just fine. Maybe I'll try adding blank comment lines and see whether it really does make it easier to spot the math. –  Mark Meckes Jul 29 '10 at 14:54
@Mark Meckes: I'm too lazy not to use VC! –  Andrew Stacey Jul 29 '10 at 18:28
@TH., @Andrew Stacey: There is another great comment in Knuth's Mathematical Writing, which says "Many readers will skim over formulas on their first reading of your exposition. Therefore, your sentences should flow smoothly when all but the simplest formulas are replaced by `\blah' or some other grunting noise." For me, this is a very good reason to always use proper punctuation at the end of displayed formulas. –  Hendrik Vogt Oct 23 '10 at 8:37