As has been repeatedly mentioned in earlier posts, contemporary American English is replete with pleonasms and other forms of linguistic hypertrophy. A well-entrenched one in speech and writing is the construction continue on, in which the postposition is superfluous, since the meaning of the verb is configured without it.
For those who are unsatisfied with a blanket resort to hypertrophy as explanans, a speculative but entirely plausible explanation of the increasing prevalence of the pleonastic variant in this case can be advanced by understanding the postposition as a syntactic diagram of the meaning of the head verb. ‘To continue’, after all, can be realized diagrammatically (iconically) through the expedient fact of its semantics being extended over linear linguistic space: to wit, the continuation of one word, beyond the simple verb, to its compound variant that contains the postposition on. This fundamental semiotic explanation has the insuperable advantage of being in complete alignment with the haecceity (quiddity) of language as a system of signs.