Light's internal reflectivity near a critical angle is very sensitive to the angle of incidence and the optical properties of the external medium near the interface. Novel applications in biology and medicine of subcritical internal reflection are being pursued. In many practical situations, the refractive index of the external medium may vary with respect to its bulk value due to different physical phenomena at surfaces. Thus, there is a pressing need to understand the effects of a refractive-index gradient at a surface for near-critical-angle reflection. In this work, we investigate theoretically the reflectivity near the critical angle at an interface with glass assuming the external medium has a continuous depth-dependent refractive index. We present graphs of the internal reflectivity as a function of the angle of incidence, which exhibit the effects of a refractive-index gradient at the interface. We analyze the behavior of the reflectivity curves before total internal reflection is achieved. Our results provide insight into how one can recognize the existence of a refractive-index gradient at the interface and shed light on the viability of characterizing it.