To continue working on something difficult without giving up.
"Learning to play the piano is hard at first, but if you keep at it, you'll improve."
To continue working hard at something despite difficulty, or to nag someone repeatedly to do something.
To not give up on a difficult task, or to keep asking someone to do something again and again.
2 meanings, ordered from most common to least. Color-coded by CEFR level.
To continue working on something difficult without giving up.
"Learning to play the piano is hard at first, but if you keep at it, you'll improve."
To repeatedly pressure or nag someone to do something.
"She kept at him until he finally agreed to call the doctor."
When used reflexively or with a task as object (keep at it / keep at the work), it means to persevere. When used with a person as object (keep at someone), it means to repeatedly push or nag that person to do something.
Natural word combinations native speakers use most often.
The five tense forms you'll use most often.
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