If is a vector valued function, prove that
is differentiable on the open interval
if and only if for all
we have
Proof. First, assume is differentiable on
. Then, by the definition of differentiability for vector valued functions we have
and each of the exists. Therefore by the usual definition of the derivative we have
Conversely, assume that
Then, the limit
exists for each . Thus,
exists for each
. Therefore,
exists