  The terms hold-in, pull-in (capture), and lock-in ranges are widely used by
engineers for the concepts of frequency deviation ranges within which PLL-based
circuits can achieve lock under various additional conditions. Usually only
non-strict definitions are given for these concepts in engineering literature.
After many years of their usage, F.~Gardner in the 2nd edition of his
well-known work, Phaselock Techniques, wrote "There is no natural way to define
exactly any unique lock-in frequency" and "despite its vague reality, lock-in
range is a useful concept." Recently these observations have led to the
following advice given in a handbook on synchronization and communications "We
recommend that you check these definitions carefully before using them." In
this survey it is shown that, from a mathematical point of view, in some cases
the hold-in and pull-in "ranges" may not be the intervals of values but a union
of intervals and thus their widely used definitions require clarification.
Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges
are given. An effective solution for the problem on the unique definition of
the lock-in frequency, posed by Gardner, is suggested.
