Inductive inference gives us a theoretical model of concept learning from examples. In this paper, we study refutably and reliably inductive inference of recursive real-valued functions. First we introduce the new criteria RealRefEx for refutable inference and RealRelEx for reliable inference. Then, we compare these two criteria with RealEx for identification in the limit, RealFin for learning finitely and RealNumi for learning by enumeration that have been already introduced in the previous works, and investigate their interaction. In particular, we show that RealRefEx and RealRelEx are closed under union, as similar as the criteria RefEx and RelEx for inductive inference of recursive functions.