The phrase "kept on" is commonly used to describe continued or persistent actions. However, there are several alternative phrases that can be used to convey a similar meaning. For example, "persisted" and "persevered" are synonyms for "kept on" that imply a dogged determination to continue despite obstacles or challenges. "Carried on" or "proceeded" are also synonyms that suggest a steady progression forward. Additionally, "maintained" or "sustained" can be used to describe a consistent effort over time. Ultimately, the choice of synonym will depend on the context, tone, and intended meaning of the sentence.