Old English ecg "corner, edge, point," also "sword" (also found in ecgplega, literally "edge play," ecghete, literally "edge hate," both used poetically for "battle"), from Proto-Germanic *agjo (source also of Old Frisian egg "edge;" Old Saxon eggia "point, edge;" Middle Dutch egghe, Dutch eg; Old Norse egg, see egg (v.); Old High German ecka, German Eck "corner"), from PIE root *ak- "be sharp, rise (out) to a point, pierce."
Spelling development of Old English -cg to Middle English -gg to Modern English -dge represents a widespread shift in pronunciation. To get the edge on (someone) is U.S. colloquial, first recorded 1911. Edge city is from Joel Garreau's 1992 book of that name. Razor's edge as a perilous narrow path translates Greek epi xyrou akmes. To be on edge "excited or irritable" is from 1872; to have (one's) teeth on edge is from late 14c., though "It is not quite clear what is the precise notion originally expressed in this phrase" [OED].
late 13c., "to give an edge to" (implied in past participle egged), from edge (n.). Intransitive meaning "to move edgeways (with the edge toward the spectator), advance slowly" is from 1620s, originally nautical. Meaning "to defeat by a narrow margin" is from 1953. The meaning "urge on, incite" (16c.) often must be a mistake for egg (v.). Related: Edger.