TITLE: What does the notation inf{...} mean?
QUESTION [8 upvotes]: I came across 
$$\inf\{k : f \in C^k\}$$
What does $\inf\{\cdot\}$ mean? I have been looking, but haven't found anything.

REPLY [4 votes]: 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.