In problem 18, enter link description here
1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by some constants, is it enough to assume $g,h$ are smooth? Otherwise, are there any examples of smooth function which are not necessarily bounded?
2) All I understood is, smoothness implies infinite differentiability, but does differentialbility guarantee boundedness, if not, any counter examples?
Thanks for any helps!
Smoothness certainly does not imply boundedness. The function $f(x) = e^x$ is smooth (analytic, even) but is unbounded along with all of its derivatives. In the link you provided the assumption is that the functions are smooth and compactly supported. The key is the last assumption. We know from real analysis that any continuous function with compact support must be bounded. Indeed, to see this we just use the extreme value theorem. Since all of the derivatives are also compactly supported, all of the derivatives are bounded.