Let T and \(T_n\) be linear bounded operators from X into Y. In this paper, we carry out the research of uniformly continuous selection of the set-valued metric generalized inverse for the first time. First, the author proves that if the 2-strictly convex space X is almost convex, N(T) is approximatively compact and R(T) is a Chebyshev hyperplane of Y, then there exists a homogeneous selection \(T^{\sigma }\) of \(T^{\partial }\) such that (1) the set \(\left( Y\backslash \overline{G_{T}}\right) \cup G_{T}\) is a dense subset of Y; (2) \(T^{\sigma }\) is continuous at every point of \(\left( Y\backslash \overline{G_{T}}\right) \cup G_{T};\) (3) \(T^{\sigma }\) is uniformly continuous on every bounded subset of \(Y\backslash \overline{G_{T}}.\) The author also proves that if X is uniformly convex and R(T) is a proximinal hyperplane of Y, then \(T^{\partial }\) is single-valued and is uniformly continuous on every bounded subset of Y. The results of this paper are to give an answer to the open problem raised by Nashed and Votruba. Finally, the author explores a perturbation problem of set-valued metric generalized inverse and proves that if X, Y are uniformly convex spaces, R(T) is a closed subspace of Y, \(N(T)\subset N(T_{n})\) for all \(n\in N\) and \(\lim _{n \rightarrow \infty }\left\| T_{n}-T\right\| =0,\) then there exists a sequence \(\{E_{k}\}_{k=1}^{\infty }\) of set such that \(T_n^\partial (y) \rightarrow {T^\partial }(y)\) uniformly on \(E_{k}\) and \({\cup }_{k=1}^{\infty }E_{k}=Y.\)
