# Proper convex function

In mathematical analysis (in particular convex analysis) and optimization, a **proper convex function** is a convex function *f* taking values in the extended real number line such that

for at least one *x* and

for every *x*. That is, a convex function is *proper* if its effective domain is nonempty and it never attains .^{[1]} Convex functions that are not proper are called *improper convex functions*.^{[2]}

A *proper concave function* is any function *g* such that is a proper convex function.

## Properties

For every proper convex function *f* on **R**^{n} there exist some *b* in **R**^{n} and β in **R** such that

for every *x*.

The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets and are non-empty convex sets in the vector space *X*, then the indicator functions and are proper convex functions, but if then is identically equal to .

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.^{[3]}