# Function is Lipschitz: $f \in C_L^{1,1}$ - what does it mean?

$$f \in C_L^{1,1}$$

I'm not sure how this set is called?

and also what are the top numbers indicating? I'm guessing one of them is how many times the function is differentiable, but what does the other number mean?

## 1 Answer

It would help if you provided context (where you found this), but without any further information, I would say it's the space of once-differentiably Hölder continuous functions that are Lipschitz. It is normal to specify the domain as well: $$C^{1,1}_L(\Omega)$$

So: if $$f\in C^{1,1}_L(\Omega)$$ then the function $$f$$ is continuous and Hölder continuous, as is its first derivative. Hölder continuity means $$|f(x)-f(y)| \leq C|x-y|^\alpha \ \forall x,y \in \Omega$$ where $$0\leq \alpha \leq 1$$ ($$1$$ in your case. If you had $$C^{1,2}_L$$ then $$0\leq \alpha \leq 2$$ for example).

If you'd like some further reference, check local Hölder continuity on page 2 of http://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pdf

• I found this in a course I'm taking in non-linear optimization. – David Refaeli Jun 18 at 15:20