"Keep on" is a commonly used phrase that refers to continuing an action or task. There are several synonyms for this phrase that can be used to add variety to writing and speech. One such synonym is "persist," which means to continue despite difficulty or opposition. Another synonym is "persevere," which adds the idea of steadfast determination and endurance. "Carry on" is a more casual alternative that is often used in colloquial speech. "Continue" and "maintain" are also frequently used synonyms that convey the idea of ongoing action. Using these synonyms can help to make writing more varied and interesting while still clearly conveying an ongoing action.