  We investigate magnetic excitations in the frustrated zigzag spin-1/2 chain
compound $\beta$-TeVO$_4$ by inelastic neutron scattering. In the magnetically
ordered ground state, the excitation spectrum exhibits coexisting magnon
dispersion, characteristic of long-range magnetic order, and a spinon-like
continuum that prevails above 2 meV, indicating the dominance of intrachain
interactions. Combining linear-spin-wave-theory and pre-calculated
spinon-continuum results, we reproduce the experimental spectrum. Our analysis
offers a minimal exchange-network model which determines dominant intrachain
interactions, their anisotropies and weak interchain interactions. The obtained
parameters explain the magnetic ordering vector and spin excitations in the
magnetic ground state.
