Rayleigh-Taylor instability (RTI) under radiation background is commonly found in both engineering applications and natural phenomena. In the optically thin and incompressible limit, the corresponding problem can be simplified as an interface discontinuous acceleration (IDA) RTI problem, but to date has only been studied in the linear stage. In this paper, the entire IDA-RTI evolution was studied numerically and theoretically, particularly for the stages beyond the linear stage. The results show that the IDA-RTI problem is equivalent to the classical RTI with the effective acceleration g_{eff}^{*} that is introduced in this work. Moreover, our studies further show that IDA-RTI can occur if and only if g_{eff}^{*}>0 (from heavy fluid to light fluid). This criterion means that IDA-RTI can occur when (i) heavy fluid supports (or accelerates) the light fluid or (ii) the two fluids have the same density, in contrast to the classical RTI problem. Moreover, the quasisteady bubble and spike velocities are theoretically predicted with quantitative accuracy, showing good agreement with the results of numerical simulations in a wide range of density ratios and acceleration configurations.