Because acceleration also varies with time, however, the extended mean value theorem must also be extended to the second time derivative to obtain the correct displacement.
By applying the mean value theorem, the operator can deduce that the vehicle's speed must equal its average speed at some time between the measurements.
If is additionally assumed to be "continuously" differentiable, then the claim can be proved by applying the mean value theorem and converting the sum into an integral.
Therefore, has to be constant on, because otherwise we would obtain a contradiction to the mean value theorem (applied separately to the real and imaginary part in the complex-valued case).