The phenomenon of bound state in the continuum (BIC) with an infinite quality factor and lifetime has emerged in recent years in photonics as a new tool to manipulate light–matter interactions. However, most of the investigated structures only support BIC resonances at very few discrete points in the ω∼k space. Even when the BIC is switched to a quasi-BIC (QBIC) resonance through perturbation, its frequency will still be located within a narrow spectral band close to that of the original BIC, restricting their applications in many fields where random or multiple input frequencies beyond the narrow band are required. In this work, we demonstrate that a new set of QBIC resonances can be supported by using a special binary grating consisting of two alternatingly aligned ridge arrays with the same period and zero-approaching ridge width difference on a slab waveguide. These QBIC resonances are distributed continuously over a broad band along a line in the ω∼k space and can thus be considered as 1D QBICs. With the Q factors generally affected by the ridge difference, it is now possible to arbitrarily choose any frequencies on the dispersion line to achieve significantly enhanced light–matter interactions, facilitating many applications where multiple input wavelengths are required; e.g., sum or difference frequency generations in nonlinear optics.