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.
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