The claim that the mathematical durationless instants do exist in a physical sense appeared always as both paradoxical and inevitable. Paradoxical, because it is difficult to find the concept of entity lasting for zero-time intelligible; inevitable, since both experience and logic suggested that it really exists. For all empirically available temporal intervals were, in the classical period at least, divisible into smaller and smaller sub-intervals unless they were durationless instants. Furthermore, the very denial of instants, as proposed by various theories of chronon, surreptitiously postulated their existence. But an attentive analysis of perceptual and, more generally, phenomenal continua shows that they consist neither of instants nor of contiguous atomic segments; their structure clearly transcends the disjunction "instants versus chronons". They exhibit a type of continuity which is different from mathematical continuity and which, as Poincaré's paradox shows, is extremely difficult to be conceptualized. There is a considerable circumstantial evidence that time on the microphysical level has a similar structure. This would make possible to deny the reality of instants without accepting the self-contradictory "atomization" of time.