Generalization of Lipschitz continuity to higher order polynomials?

Question

Lipschitz continuity of a function $x\to f(x) $ is defined as: $$\exists K,\forall x_1,x_2 : \frac{|f(x_1)-f(x_2)|}{|x_1-x_2|}\leq K$$

This can be viewed with pairs of lines bounding the function in the graph

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A line is a first order polynomial. Does there exist any higher order polynomials which we can define other kinds of continuity with? Would there be anything to gain from trying to define it? For example we could imagine second order polynomial (parabola) bounding at each point instead of lines. Something like this (?)

$$\exists K,\forall x_1,x_2: \frac{|f(x_1)-f(x_2)|}{(x_1-x_2)^2}\leq K$$

Answer 1

If $f$ is differentiable you can consider Lipschitz continuity of its derivative. If $f'(x)$ is Lipschitz continuous with constant $L$, then it is true that $$\forall x,y \in\mathbb{R}\quad f(x)+f'(x)(y-x)-\frac{L}{2}|y-x|^2\leqslant f(y)\leqslant f(x)+f'(x)(y-x)+\frac{L}{2}|y-x|^2,$$

so you have global quadratic bounds. Similarly, Lipschitz continuity of higher-order derivatives implies polynomial bounds of higher degree. The same holds true in the multivariate case, which is heavily used in numerical optimization.

Answer 2

If : $$ \frac{|f(x_1)-f(x_2)|}{(x_1-x_2)^2}\leq K$$

Then :

$$\frac{|f(x_1)-f(x_2)|}{|x_1-x_2|}\leq K |x_1-x_2| \underset{x_2 \to x_1}\to 0$$

Then $f$ has a derivative, which is zero everywhere, thus $f$ is a constant function.

Note that this remains true if you replace $2$ by any $\alpha>1$.