For a given first category subset E of the unit
circle and any given holomorphic function g on the open unit
disk, we construct a universal Taylor series f on the open unit
disk, such that, for every n = 0,1,2,..., f(n) is close to
g(n) on a set of radii having endpoints in E. Therefore,
there is a universal Taylor series f, such that f and all its
derivatives have radial limits on all radii with
... [Show full abstract] endpoints in E.
On the other hand, we prove that if f is a universal Taylor
series on the open unit disk, then there exists a residual set G
of the unit circle, such that for every strictly positive integer
n, the derivative f(n) is unbounded on all radii with
endpoints in the set G.