## Can a discontinuous function be convex?

Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable. If, for some function f, inequality (2) is true for any two points x1 and x2 in some interval and any p1>0 and p2>0, the function f is continuous and, of course, convex on this interval.

**Is a single point a convex set?**

The empty set ∅, a single point {x}, and all of Rn are all convex sets.

### What is convex down?

Quick Reference. Some authors say that a curve is convex up when it is concave down, and convex down when it is concave up (see concavity).

**What is concave and convex?**

Concave means “hollowed out or rounded inward” and is easily remembered because these surfaces “cave” in. The opposite is convex meaning “curved or rounded outward.” Both words have been around for centuries but are often mixed up.

## Is a circle convex?

The interiors of circles and of all regular polygons are convex, but a circle itself is not because every segment joining two points on the circle contains points that are not on the circle. To prove that a set is convex, one must show that no such triple exists.

**Is concave or convex positive?**

focal length | object distance | |
---|---|---|

concave mirror | positive | positive |

convex mirror | negative | positive |

converging lens | positive | positive |

diverging lens | negative | positive |

### How do you know if its 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.

**How to prove a function is convex?**

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.) Proof: This is straightforward from the definition.

## Is a convex function Lipschitz continuous?

Convex functions are Lipschitz continuous on any closed subinterval. Strictly convex functions can have a countable number of non-differentiable points. Eg: f(x) = ex if x < 0 and f(x)=2ex − 1 if x ≥ 0. So max{ex,e−x} is strictly convex and not differentiable at 0.

**Is R2 a convex set?**

Intuitively if we think of R2 or R3, a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next figure). Here is the definition. In, say, R2 or R3, this set is exactly the line segment joining the two points u and v. (See the examples below.)

### What is convex set and non convex set?

A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point. A set C is absolutely convex if it is convex and balanced.

**What is a convex lens?**

What is Convex Lens? The convex lens is a lens that converges rays of light that convey parallel to its principal axis (i.e. converges the incident rays towards the principal axis) which is relatively thick across the middle and thin at the lower and upper edges. The edges are curved outward rather than inward.

## What is the difference between Plano-convex and concave lenses?

However, if one of the surfaces is flat and the other convex, then it is called a plano-convex lens. There is another type of lens known as concave lens. The major differences between concave and convex lens two are:

**Why are the edges of convex lenses curved outward?**

The edges are curved outward rather than inward. It is used in front of the eye bends the incoming light sharply so the focal point shortens and the light focuses properly on the retina. Generally, a convex lens can converge a beam of parallel rays to a point on the other side of the lens.

### Which function is lower semicontinuous at ˉx?

Since this is also true for x = ˉ x, the function f is lower semicontinuous at ˉ x. Let f: D → R and let ˉx ∈ D.