Time Complexity of Regular Languages
This is a classical result of Kobayashi, On the structure of one-tape nondeterministic Turing machine time hierarchy, Theorem 3.3 on page 188. The idea is to use crossing sequences: you show an upper bound $O(1)$ on the size of any crossing sequence, and then use an argument of Hennie, One-tape off-line Turing machine computations, Theorem 2 on page 561, to conclude that the language accepted by the machine is regular.