There are several synonyms for the phrase "continue on." One of them is "persist," which means to continue to exist or endure. Another synonym is "proceed," which means to continue to go forward. "Carry on" and "keep going" are also frequently used to refer to continuing an action or journey. "Press on" is a more forceful way to say "continue on," implying that one should push through obstacles or difficulty. "Forge ahead" is another similar term that emphasizes a sense of determination or courage. All of these synonyms can be used interchangeably with "continue on" depending on the context and tone of the sentence.