When to begin a new paragraph? When writing a mathematical article, when should someone begin a new paragraph? Is there some specific rule or convention?
And more generally, what rules are there about articles-writing?
 A: Mathematical writing, like all writing, is best when it reads smoothly.  A good way to test the smoothness of any piece of writing is to read it out loud to yourself (I realize that this test works best for those mathematicians who write in their native languages).  Sentence and paragraph breaks should be at natural points in the flow of an argument.  Certainly you should (at minimum) start a new paragraph any time you move from one argument (lemma, definition, remark) to another.
There are special considerations, of course, when writing in any specific discipline.  It is rare for mathematicians to read an article straight through, at least on the first pass, so it is very important that your writing is skimmable.  Paragraph breaks and other formatting are one of the key ways to achieve this.  Short, precise sentences, in subject-verb-object order, are preferable to long convoluted sentences, even when it makes the text sound slightly choppy.  (I often fail at my own advice, and produce mathematical writing that is too wordy, with long blocks of text.)
Actually, paragraph breaks are just the beginning: you should clearly label each block of a few paragraphs for what it is (a definition, a remark, a proof), and headers can be very helpful ("Definition (gizmo): A gizmo is a thingamabob equipped with a compatible doohickey.").  It benefits the reader if you indent and italicize (and so on) the main theorems and definitions.
Of course, these things also depend in part on the venue.  Writing for math.stackexchange is slightly different from writing an article (although still definitely "mathematical writing").  Writing reviews is again different.
As was mentioned in the comments, your question here has perfect paragraph breaks.
In fact, glancing over your other questions on m.se, it looks like your paragraph breaks are perfectly placed.
A: Someone previously posted a link to this excellent style guide by Knuth and others. It is worth reading all of it.
Some inexperienced writers have an awkward way of mixing equations and prose. Instead of putting the displayed formula
$$f(x)=\ldots$$
in the sentence, they break the flow by ending the sentence just before the equation and then beginning a new sentence.
If you ever write for non-mathematicians (such as me), it may be fine to have the same amount of equations as you would when writing for a professional, if you just add more prose around the formulas that help explain them. It is the equation density per page that matters. There was a paper on that topic in PNAS that actually stirred some debate.
