at a macroscopical level, things appear to be a wave (continuity). but at a microscopical level, things appear to a particle (discontinuity).
consider the Young split experiment. based on the result of the firing of just one photon, u will notice that it hits the detector at exactly ONLY one place. its only when a second photon comes, that amazingly, appears on the spot where a constructive interference pattern would be if a wave was transmitted rather. well not exactly, but the more photons you fire, the more likely it appears as a wave pattern with constructive and destructive interference.
therefore, to me, there is no problem distinguising the two, it is just from the context you look at, which is macroscopical or microscopical.