No, inline equations are not split (fortunately). While inline math equations arguably appear more frequently, display math equations (as the big equations are called in TeX terms) are pretty frequent, too.
It is bad form to let equations stand on their own and not embed them into full sentences. One finds almost never exceptions to this in the literature.
I think the best way would be to just not treat “$$” (the delimiters of display math) as a delimiter of blocks just as “$” is also not treated as a delimiter of blocks.
Then,
We are interested in the set
$${x\in\mathbb{R}\colon a\leq x\leq b}$$
which is just the interval $[a,b]$ containing all numbers between $a$ and $b$.
would be a single block (as it should be), while
Euler’s identity:
$$e^{i\pi} = -1$$
Continuing, …
would make the equation a distinct block as could sometimes be intended.
Display math mode should be used whenever printing the formula centered and big makes things more readable. Nonetheless, the mathematical content is still part of its enclosing paragraph whether or not it is printed inline or big.
Thank you for your patience!