real analysis - Does there exist a function such that the following criteria are satisfied simultaneously? Can you provide an example?

Does there exist a function such that the following criteria are satisfied simultaneously?

$\bullet$ It and all of its derivatives up to any order have compact support

$\bullet$ Has an unbounded derivative (at some order)

$\bullet$ Has an everywhere defined derivative for all orders

I thought that at least one of the derivatives of the function would have to be discontinuous since all of its derivatives have compact support and so continuity would imply boundedness (extream value theorem). I couldn't come up with an example however. Can anyone provide an example if its true?

Answer

This is impossible: as you say, by the extreme value theorem one of the derivatives would have to be discontinuous. But the derivatives are all differentiable (since the next derivative exists) and hence continuous.