  In this article, we prove the existence of solutions to a nonlinear nonlocal
elliptic problem with a singualrity and a discontinuous critical nonlinearity
which is given as follows.
  \begin{align}
  \begin{split}\label{main_prob}
  (-\Delta)_p^su&=\mu
g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega
  u&>0,~\text{in}~\Omega,
  u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega,
  \end{split}
  \end{align}
  where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz
boundary, $s\in (0,1)$, $2<p<\frac{N}{s}$, $\gamma\in (0,1)$, $\lambda,\mu>0$,
$\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq
0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical
Sobolev exponent.
  Under suitable assumptions on the function $g$, we prove the existence of
solution to the problem. Furthermore, we show that as $\alpha\rightarrow0^+$,
the sequence of solutions of $\eqref{main_prob}$ for each such $\alpha$
converges to a solution of the problem for which $\alpha=0$.
