derivative of limit function diverges from limit of derivatives
For a function sequence, one cannot always change the of http://planetmath.org/node/6209taking limit and differentiating (http://planetmath.org/Differentiate), i.e. it may well be
in the case that a sequence of continuous (and differentiable
) functions converges uniformly; cf. Theorem 2 of the parent entry (http://planetmath.org/LimitFunctionOfSequence).
Example. The function sequence
(1) |
provides an instance; we consider it on the interval . It’s a question of partial sum the converging geometric series
(although one cannot use Weierstrass’ criterion of uniform convergence). Sincethe limit function is
we have
which means by Theorem 1 of the parent entry (http://planetmath.org/LimitFunction) that the sequence (1) converges uniformly on the interval to the identity function. Further, the members of the sequence are continuous and differentiable. Furthermore,
whence
But in the point we have
which says that the limit of derivative sequence of (1) is discontinuous
in the origin. Because
we may write