What does the notation inf{...} mean? I came across 
$$\inf\{k : f \in C^k\}$$
What does $\inf\{\cdot\}$ mean? I have been looking, but haven't found anything.
 A: It means infimum. So, $\inf\{\ldots\}$ is the infimum of the set $\{\ldots\}$ (assuming that it is a non-empty set of real numbers with a lower bound).
A: Lower bound of some set of numbers is number which smaller or equal to any  number of the set.
Greatest lower bound of some set of numbers is a number which is a lower bound of the set and is bigger or equal to any other lower bound of the set. $\inf A$ means greatest lower bound of the set $A$. So e.g. $\inf A = 5$ means greatest lower bound of the set $A$ is $5$.
Greatest lower bound is also called infimum.
A: Suppose that you have a non-empty set of numbers,as an example $A = \{1,10,\pi,55, 11.2, \sqrt{2}, {1\over 2}\}$ then the infimum of this set is the greatest lower bound of the set. In this simple case $$\inf\{A\}={1\over 2}$$
As an added bonus: what you gave us $$\inf\{k: f\in C^k\}$$ means that, given a function $f$ differentiable $n$ times, the infimum is the lower possible $k$ such that the $k$-th derivative of this function is continuous. But I think that $k$ will be $0$ every time..
