I have been reading here for several days, and the definition of boundedness is that there is a positive integer m, which makes all objects in the domain defined.

丨丨 f(x) ≤M, and "the n-order derivative at f(x) is bounded in the interval (a, b)" means n- 1 order n-2. . . The derivative function is derivable, so the derivative can always be obtained. Assuming that the first derivative is not differentiable, then the n derivative cannot be obtained;

"f(x) has a derivative up to N order in the interval (a, b) containing x0, and the N order derivative of f(x) is continuous in (a, b)", which also means that the N order derivative of f(x) exists and is bounded. If the n derivative of f(x) is unbounded, it means that the derivative function of n- 1 is not derivable.

So the above two conditions are consistent.

It doesn't seem to make sense. . .