Middle English kēpen, from late Old English cepan (past tense cepte) "to seize, hold; seek after, desire," also "to observe or carry out in practice; look out for, regard, pay attention to," from Proto-Germanic *kopjan, which is of uncertain origen. Old English cepan was used c. 1000 to render Latin observare, so perhaps it is related to Old English capian "to look" (from Proto-Germanic *kap-), which would make the basic sense "to keep an eye on, see to it."

The word prob. belonged primarily to the vulgar and non-literary stratum of the language; but it comes up suddenly into literary use c. 1000, and that in many senses, indicating considerable previous development. [OED]

The senses exploded in Middle English: "to guard, defend" (12c.); "restrain (someone) from doing something" (early 13c.); "take care of, look after; protect or preserve (someone or something) from harm, damage, etc." (mid-13c.); "preserve, maintain, carry on" a shop, store, etc. (mid-14c.); "prevent from entering or leaving, force to remain or stay" (late 14c.); "preserve (something) without loss or change," also "not divulge" a secret, private information, etc., also "to last without spoiling" (late 14c.); "continue on" (a course, road, etc.), "adhere to" a course of action (late 14c.); "stay or remain" (early 15c.); "to continue" (doing something) (mid-15c.). It is used to translate both Latin conservare "preserve, keep safe" and tenere "to keep, retain."

From 1540s as "maintain for ready use;" 1706 as "have habitually in stock for sale." Meaning "financially support and privately control" (usually in reference to mistresses) is from 1540s; meaning "maintain in proper order" (of books, accounts) is from 1550s.

To keep at "work persistently" is from 1825; to keep on "continue, persist" is from 1580s. To keep up is from 1630s as "continue alongside, proceed in pace with," 1660s as "maintain in good order or condition, retain, preserve," 1680s as "support, hold in an existing state." To keep it up "continue (something) vigorously" is from 1752. To keep to "restrict (oneself) to" is from 1711. To keep off (trans.) "hinder from approach or attack" is from 1540s; to keep out (trans.) "prevent from entering" is from early 15c.

c. 1300, sauf, "unscathed, unhurt, uninjured; free from danger or molestation, in safety, secure; saved spiritually, redeemed, not damned;" from Old French sauf "protected, watched-over; assured of salvation," from Latin salvus "uninjured, in good health, safe," which is related to salus "good health," saluber "healthful" (all from PIE *solwos from root *sol- "whole, well-kept"). For the phonological development of safe from sauf, OED compares gage from Old North French gauge.

From late 14c. as "rescued, delivered; protected; left alive, unkilled." The meaning "not exposed to danger" (of places, later of valuables) is attested from late 14c.; in reference to actions, etc., the meaning "free from risk," is recorded by 1580s. The sense of "sure, reliable, not a danger" is from c. 1600. The sense of "conservative, cautious" is from 1823. It has been paired alliteratively with sound (adj.) from c. 1300. In Middle English it also meant "in good health," also "delivered from sin or damnation." Related: Safeness.