Synonyms for Keep up:

n.

arise, retire, lie in, rise, early bird, abed, get up, lay up, sit-up. safeguard, absorb, train, pick up, learn, retrain, acquire, care for, develop, master, acquaint yourself with something. pretend, simulate, fake, tease, persist, play out, persevere. progress, forge ahead, advance, compete, get on with, get ahead, reach, get along, equal, break through, rival, plod along, outpace. fall/fit into place, see the light, get the idea, deduce, make sense of something, have a feel for something, know, work out, understand, comprehend. follow through, press ahead, stay the course, go on, keep on, get on. DO, leave behind, travel. keep up (noun)
preserve, prolong, conserve, sustain, keep abreast, maintain, follow.

v.

maintain, sustain (verb)
rival, go on, compete, preserve, persevere.

Other synonyms:

tease, master, rise, retire, lie in, pretend, retrain, get up, abed, play out, pick up, fake, train, early bird, develop, acquire, lay up, sit-up, learn. simulate. absorb. leave behind. travel. DO. continue
forge ahead, go on, keep on.
support
stay the course.
Other relevant words:
persevere, persist, care for, rival, go on, keep on, compete.