Surface-brightness profiles (left), intrinsic luminosity density profiles (middle), and logarithmic density slopes (right) for sharp-transition core–Sérsic models (α → ∞). All models have a total luminosity of L = 1, a break radius of R_b = 1, and an effective radius of R_e = 5. In the top row the Sérsic index is fixed to m = 4, while the inner power-law slope, γ, is varied as indicated; in the bottom row the inner slope is fixed to γ = 0.25 and the Sérsic index m is varied. The projected profiles show the expected behaviour of a pure power law for R ≤ R_b joined to a Sérsic profile for R ≥ R_b, with a discontinuity in the radial derivative at R_b. This discontinuity translates into a sharp peak in the intrinsic density at r ≈ R_b and a region with γ(r) < 0 just inside the break radius, demonstrating that all sharp-transition core–Sérsic models have non-monotonic and hence unphysical intrinsic density profiles.