This article develops joint inferential methods for the cause-specific hazard function and the cumulative incidence function of a specific type of failure to assess the effects of a variable on the time to the type of failure of interest in the presence of competing risks. Joint inference for the two functions are needed in practice because (i) they describe different characteristics of a given type of failure, (ii) they do not uniquely determine each other, and (iii) the effects of a variable on the two functions can be different and one often does not know which effects are to be expected. We study both the group comparison problem and the regression problem. We also discuss joint inference for other related functions. Our simulation shows that our joint tests can be considerably more powerful than the Bonferroni method, which has important practical implications to the analysis and design of clinical studies with competing risks data. We illustrate our method using a Hodgkin disease data and a lymphoma data. Supplementary materials for this article are available online.