why is the energy distribution of beta rays continuous?

Because along with the beta particle an anti neutrino is also emitted in beta decay and also they share the total energy in all possible proprtions So the energy distribution of beta rays continuous.

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the energy of beta rays is continuous due to emmision of neutrino.

In fact ifneutrino would not be emitted,the energy distribution would be discontinuous.

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When a nucleus beta-decays, it emits two particles at the same time: One is a beta particle; the other, a neutrino or antineutrino. With this emission, the nucleus itself undergoes a transformation, changing from one element to another. In the case of isotopes that β+-decay, each decaying nucleus emits a positron and a neutrino, simultaneously reducing its atomic number by one unit; for those isotopes that β-decay, each nucleus emits an electron and an antineutrino while increasing its atomic number by one. In both classes of decay, the energy released by the nuclear transformation is shared between the two emitted particles. Though the energy released by a particular nuclear transformation is always the same, the fraction of this energy carried away by the beta particle is different for each decaying nucleus. (The neutrino always carries away the remainder, thus conserving energy overall.) When observed collectively, the decaying nuclei of a simple radioactive source emit their beta particles with a continuous distribution of kinetic energies covering the range from zero up to the total nuclear decay energy available.
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Bet the next person viewing this answer is an Apeejayite too !! Cheers ;)
 
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