to keep up with somebody:
to write to, telephone, or meet a friend regularly, so that you do not forget each other

a través de keep up with somebody | meaning of keep up with somebody in Longman Dictionary of Contemporary English | LDOCE

but

to keep something ↔ up: to continue doing something
“I don’t think I can keep this up any longer”
“keep up the good work!” (=continue to work hard and well)

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