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.

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.
