The number of directions determined by a function over a finite field

A proof is presented that shows that the number of directions determined by a function over a finite field GF(q) is either 1, at least (q+3)/2, or between q/s+1 and (q−1)/(s−1) for some s where GF(s) is a subfield of GF(q). Moreover, the graph of those functions that determine less than half the directions is GF(s)-linear. This completes the unresolved cases s=2 and 3 of the main theorem in Blokhuis et al. (J. Combin. Theory Ser. A 86 (1999) 187).