There is an important difference between the paradox of the heaps and the fallacy of the heaps. The former also is known as the continuum paradox or the sorites paradox (from σωρείτης), and the latter also is known as the continuum fallacy or sorites fallacy. (soros, from σωρός, is Greek for "heap".)
The paradox of the heaps is the paradox described in the prompt-question of this thread. A man was once a boy, but the instant the boy became a man is equivocal. The continuum of a lifetime seems to make arbitrary any rigid designations of time distinguishing childhood from adulthood. Nonetheless, it would be fallacious to argue that this, on its own, implies that a lifetime either is entirely childhood or is entirely adulthood.
It is a fallacy of the heaps to argue that the fact of continuum precludes any meaningful distinctions of regions along a continuum. For example, amounts of heat vary on a continuum, and water boils/freezes when it becomes sufficiently hot/cold such that, on a heat-continuum, there are meaningfully distinct regions. Therefore, it is not true that, for every continuum, there are no meaningfully distinct regions. Of course, there are some continuums in which there do not seem to be any meaningfully distinct regions. For example, the real number line, as a continuum of 1-dimensional space, does not seem to have any meaningful distinct regions, although it has meaningfully distinct subsets (e.g. the set of integers, of irrational numbers, and of natural numbers).
