Summary: For the singular integral operators with piecewise continuous coefficients the authors prove the criterion of Fredholmness and formula for the index in the generalized Lebesgue spaces \(L^{p(\cdot)}(\Gamma)\) on a finite closed Lyapunov curve \(\Gamma\) or a curve of bounded rotation. The obtained criterion shows that Fredholmness in this space and the index depend on values of the function \(p(t)\) at the discontinuity points of the coefficients of the operator, but do not depend on values of \(p(t)\) at points of their continuity.