Dear MHF members,
the problem is the following.
Show that the functions
are of class .
Here I will only show that is of class since the same arguments apply for too.
Let and , then by the mean value theorem, we have , where is between and , which yields Thus, for all , we have since . Given , we may let to have for all with . This shows that is continuous.
For the next step, we shall show that for , the integral converges uniformly, i.e., for every there exists such that for all and .
Since , for any given , there exits such that , which yields for all . Therefore, the integral converges uniformly, which allows us to change the order of the differential operator and the integral. Then, for all , we have showing that exists. We can show that is continuous as well, i.e., is in .