Thus Δ Δ 1 t Δ Δ 2 t f ( x 0 , y 0 ) tends to D 1 D 2 f ( x 0 , y 0 ) as t tends to 0. The same argument shows that Δ Δ 2 t Δ Δ 1 t f ( x 0 , y 0 ) tends to D 2 D 1 f ( x 0 , y 0 ) . Hence, since the difference operators commute, so do the partial differential operators D 1 and D 2 , as claimed.