keep on
keep on {v.}
1. To go ahead; not stop; continue.
The neighbors asked them to stop making noise, but they kept right on.
Columbus kept on until he saw land.
- Often used before a present participle.
Relentlessly, the boy kept on asking about the birds and the bees.
The boy kept on talking even though the teacher had asked him to stop.
2. To allow to continue working for you.
The new owner kept Fred on as gardener.
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Source: A Dictionary of American Idioms