The purpose of this article is to prove the existence of solution to a nonlinear nonlocal elliptic problem with a singularity and a discontinuous critical nonlinearity, which is given as (−Δ)psu= μg(x,u)+λuγ+H(u−α)ups*−1inΩ,u>0inΩ, with the zero Dirichlet boundary condition. Here, Ω⊂RN is a bounded domain with Lipschitz boundary, s ∈ (0, 1), 2<p<Ns, γ ∈ (0, 1), λ, μ > 0, α ≥ 0 is real, ps*=NpN−sp is the fractional critical Sobolev exponent, and H is the Heaviside function, i.e., H(a) = 0 if a ≤ 0 and H(a) = 1 if a > 0. Under suitable assumptions on the function g, the existence of solution to the problem has been established. Furthermore, it will be shown that as α → 0+, the sequence of solutions of the problem for each such α converges to a solution of the problem for which α = 0.