What is the correct term for the maximal running time of a given algorithm on all inputs of length bounded by given $n$, on which the algorithm halts?  Assume, if necessary, that the halting problem of the given algorithm is decidable (with unknown complexity).How is this time function of $n$ called?  Is it studied in complexity theory, or does one only consider the running time of universally halting algorithms?
Running time of partial algorithms
complexity theory;terminology;reference request;time complexity