I am beginner when it comes to convex optimization and had to read a part Lecture notes of Prof Nesterov on Convex Optimization for my class (needless to say I wasn't too thrilled). I had a doubt in the explaination of classes of differentiable function given on page 20.
It says that a class of functions given by $C_L^{k,p}(Q)$ has the following properties:
$ f \in C_L^{k,p}(Q)$ is $k$ times continuously differentiable on $Q$
The $p$th derivative is Lipschitz continuous on $Q$ with the constant $L$: $$\|f^{(p)}(x)-f^{(p)}(y)\| \leq L\|x-y\|$$ for all $x,y \in Q$
Then it states two statements that I didn't understand:
1) And clearly, we always have $p\leq k$
2) If $q \geq k$ then $C_L^{q,p}(Q) \subseteq C_L^{k,p}(Q)$
Could somebody please explain why the following two hold true? How would a function not be Lipschitz continuous but still have higher derivatives? How does the second statement hold?