Let x , e ⩾ 0 , u 0 > ⋯ > u d + e x,e \geqslant 0,{u_0} > \cdots > {u_{d + e}} and h > 0 h > 0 be real numbers. Let f f be a real valued function and let Δ ( h ; u , w ) f ( x ) h − d \Delta (h;u,w)f(x){h^{ - d}} be a difference quotient associated with a generalized Riemann derivative. Set I = ( x + u 0 h , x + u d + e h ) I = (x + {u_0}h,x + {u_{d + e}}h) and let f f have its ordinary ( d − 1 ) (d - 1) st derivative continuous on the closure of I I and its d d th ordinary derivative f ( d ) {f^{(d)}} existent on I I . A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a ξ ∈ I \xi \in I such that the difference quotient is equal to f ( d ) ( ξ ) ) {f^{(d)}}(\xi )) is given for d = 1 d = 1 and d = 2 d = 2 . The condition is sufficient for all d d . It is used to show that many generalized Riemann derivatives that are "good" for numerical analysis do not satisfy this mean value theorem.