Is beta decay continuous?
In my opinion, yes and no.
You might want to elaborate on the question. Anyway, when we talk about their energy (kinetic energy), beta decay is continuous. Scientists found out that the electrons they detected have a range of energies. The discovery was unexpected because they would think that they would get a constant energy for beta decay.
If we disregard the energy, beta decay is still both continuous and not because of the wave-particle duality .
Discrete because beta decay is essentially a release of particles by radioactive atoms . They behave like particles when they interact with other particles (eg: air, other material).
Continuous because beta decay can show wave properties under certain conditions (eg: Young's double slit experiment)