From AMS Glossary
In its restricted physical sense, any physical quantity that varies in three-dimensional space (and possibly time), usually continuously except possibly on surfaces or curves.
Field quantities often satisfy partial differential equations. An example of a scalar field is the temperature T(x, y, z, t) at time t at each point (x, y, z) of a solid body; an example of a vector field is the (local) velocity field v(x, y, z, t) in a fluid, the separate parts of which are in motion relative to each other. The continuity of these fields is a mathematical fiction, obtained by averaging over volumes containing many atoms or molecules but still small on a macroscopic scale.