DearMHFmembers,

the problem is the following.

Show that the functions

and

are of class .

Thanks.

bkarpuz

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- May 2nd 2011, 09:20 PMbkarpuzFunctions of class $C^{1}$?
Dear

**MHF**members,

the problem is the following.

Show that the functions

and

are of class .

Thanks.

**bkarpuz** - May 7th 2011, 12:34 PMbkarpuz
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 . - May 7th 2011, 11:44 PMJose27
How about this: Call the integrand for , then and where are integrable, and using the mean value theorem and dominated convergence we obtain the result.