Characterizations of ITBM-computability. I

We examine whether some well-known properties of partial recursive functions are valid for ITBM-computable functions, i.e., functions which can be computed with infinite time Blum–Shub–Smale machines. It is shown that properties of graphs of ITBM-computable functions differ from the usual properties of graphs of partial recursive functions. All possible ranges of ITBM-computable functions are described, and we consider the existence problem for ITBM-computable bijections between ITBM-computable sets of the same cardinality.