Can a non continuous function be convex?

There exist convex functions which are not continuous, but they are very irregular: If a function f is convex on the interval (a,b) and is bounded from above on some interval lying inside (a,b), it is continuous on (a,b). Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.

Are convex function always differentiable?

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable.

How do you prove a convex function is continuous?

Secondly, we prove that any real-valued convex function on an open convex set is lo- cally Lipschitzian and hence continuous. Hence if the interior of the domain of a convex function f is not empty, then f is continuous on the interior of its domain.

Can a discontinuous function be concave?

A concave function can be discontinuous only at an endpoint of the interval of definition.

Does a convex function have a unique minimum?

It is well-known that if a convex function has a minimum, then that minimum is global. The minimizers, however, may not be unique. There are certain subclasses, such as strictly convex functions, that do have unique minimizers when the minimum exists, but other subclasses, such as constant functions, that do not.

How do you know if a function is convex?

Theorem 1. A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.)

Does a strictly convex function have a minimum?

If f is strictly convex, then there exists at most one local minimum of f in X. Consequently, if it exists it is the unique global minimum of f in X.

How do I know if a function is convex?

Concave and convex functions. Let f be a function of many variables, defined on a convex set S. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph; f is convex if the line segment joining any two points on the graph is never below the graph.

Is the twice differentiable convex function strictly convex?

Visually, a twice differentiable convex function “curves up”, without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold.

How is the assumption of convexity related to quasiconvexity?

Simply apply the definition of a concave function. An identical result holds for convex functions. The assumption of convexity has two important implications. First, every concave function must also be continuous except possible at the boundary points. 3 Second, every concave function is differentiable “almost everywhere”. Theorem 4.

When is a convex function called a quasiconvex function?

A strictly convex function will have at most one global minimum. the sublevel sets { x | f ( x) < a } and { x | f ( x) ≤ a } with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function. .

Which is easier to work with convex or strongly convex functions?

Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets. {\\displaystyle \\phi } is a function that is non-negative and vanishes only at 0.