To prove that at

is it sufficient to show that and are continous for in the range of integration and for in a interval about ? And if you want to differentiate under the integral sign multiple times, is it sufficient to show that each derivative of is continous for in the range of intergration and for in a interval about ?

And what about improper integral ?

Is it sufficient to show in addition that and are bounded above by functions independent of and the integrals of both of those functions converge over the interval [a, ?