Conditions under which the power curve is not smooth or continuous in Hypothesis testing

Question

Not sure if this is a valid question.

I was wondering if there is are good answers to this question:

Under what conditions is the power curve not smooth or continuous in Hypothesis testing?

Thanks.

Answer 1

When the test involved is a permutation test the number of possible permutations is limited and so the power curve has to have steps.

Answer 2

Where the parameter-space is discontinuous?

e.g. Hypothesis test of a parameter for a hypergeometric.