Can a discontinuous function be convex?

Suppose that I is a closed interval: I=[a,b] with aeasily verified that f is convex. However, f is not continuous at 0 and 1.

Are continuous functions absolutely continuous?

If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous. Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.

How do you know if a function is absolutely continuous?

Theorem 1.2. If f is absolutely continuous on [a, b] and f (x) = 0 for almost every x ∈ [a, b], then f is constant. |f(cx) − f(ax)| < ε(cx − ax).

Is a concave function continuous?

This alternative proof that a concave function is continuous on the relative interior of its domain first shows that it is bounded on small open sets, then from boundedness and concavity, derives continuity. If f : C → R is concave, C ⊂ Rl convex with non-empty interior, then f is continuous on int(C).

Is it possible for a concave function to have discontinuities?

Concavity can change at a discontinuity, but it won’t be an inflection point.

Is absolutely continuous with respect to?

A concept in measure theory (see also Absolute continuity). If μ and ν are two measures on a σ-algebra B of subsets of X, we say that ν is absolutely continuous with respect to μ if ν(A)=0 for any A∈B such that μ(A)=0 (cp.

Is Cantor function absolutely continuous?

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).

What is absolutely continuous random variable?

A random variable is absolutely continuous iff every set of measure zero has zero probability. For this reason, it is called a measure, but not a probability measure. 8. Any finite or countable infinite set has measure zero. There are also some uncountable sets that have measure zero.

Is every convex function is continuous?

Since in general convex functions are not continuous nor are they necessarily continuous when defined on open sets in topological vector spaces. But every convex function on the reals is lower semicontinuous on the relative interior of its effective domain, which equals the domain of definition in this case.

How do you tell if a function is convex or concave?

To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave. To find the second derivative, we repeat the process using as our expression.

Is a convex function always continuous?

All measurable convex functions on open intervals are continuous. 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).

How do you determine if a function is convex or concave?

To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.

Is the convex function f ( x ) a continuous function?

Since in general convex functions are not continuous nor are they necessarily continuous when defined on open sets in topological vector spaces. An alternative might be to identify the point of discontinuity as x. Then there exists a point arbitrarily close to x, denoted x’, whose value f (x’) is bounded away by a constant from f (x).

Is the absolute value function of one variable convex?

Functions of one variable. The absolute value function is convex (as reflected in the triangle inequality ), even though it does not have a derivative at the point x = 0. It is not strictly convex.

Which is true of every convex function on the reals?

But every convex function on the reals is lower semicontinuous on the relative interior of its effective domain, which equals the domain of definition in this case. A more general proof of this property is given in “Convexity and Optimization in Banach Spaces.”

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.

Can a discontinuous function be convex? Suppose that I is a closed interval: I=[a,b] with aeasily verified that f is convex. However, f is not continuous at 0 and 1. Are continuous functions absolutely continuous? If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous. Every absolutely…