This is exercise 4.8.6 from Hils and Loeser's A First Journey through Logic: Prove that for every Turing machine $\mathcal M$ which computes a total function, the graph of the computing time function $T_{\mathcal M}$ is primitive recursive.
Earlier in the text, we have a construction of the primitive recursive function Sit($t$). This returns the "situation" of $\mathcal M$ at time $t$, which is the current state, the position of the machine head, and the tape configuration. Clearly I'd like to find the least time at which the state is the final state. However, in order for this to be built as a primitive recursive function, the quantification must be bounded. However, I can't think of any bound which is guaranteed to be greater than the computing time of a machine on a given input.