# 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.

• It means infimum. Jul 21, 2018 at 11:16
• It doesn't make a lot of sense in this particular context, since if $f$ is $\mathcal C^k$ for any one $k$, then it is also $\mathcal C^0$, so this infimum is either $0$ or (perhaps) $\infty$. Jul 21, 2018 at 11:23

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 infinium 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..

• "lower upper bound"?? Jul 21, 2018 at 11:24
• Oops, my bad, I'm correcting it Jul 21, 2018 at 11:24
• Maybe you meant not "infinium" but "infimum" and not "lower bound" but "greatest lower bound" Jul 21, 2018 at 11:35
• Thank you all, I wrote the answer quickly! Now it's all corrected Jul 21, 2018 at 11:37
• And by axiom, every non empty down bounded set of real numbers has infimum... Jul 21, 2018 at 12:17

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).

• To save a google search: en.m.wikipedia.org/wiki/Infimum_and_supremum Jul 21, 2018 at 11:19
• This answer is a bit like Q “what does etc. mean” being answered with “etc. means et cetera.” I.e. factually correct but not really what the OP was seeking. If the OP knew what infimum was they would know what inf meant. Sep 21, 2022 at 21:02

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.