The exponentiation formula of reliability and survival: does it always hold?

In reliability and biometry, it is common practice to choose a failure model by first assessing the failure rate function subjectively, and then invoking the well known "exponentiation formula". The derivation of this formula is based on the assumption that the underlying failure distribution be absolutely continuous. Thus, implicit in the above approach is the understanding that the selected failure distribution will be absolutely continuous. The purpose of this note is to point out that the absolute continuity may fail when the failure rate is assessed conditionally, and in particular when it is conditioned on certain types of covariates, called "internal covariates". When such is the case, the exponentiation formula should not be used.