Discontinuous convex functions exist, but they are very irregular: if a function f is convex on the interval (a,b) and bounded from above on some interval inside (a,b), then it is in ( a, b).Therefore, a discontinuous convex function unbounded and unmeasurable over any interior interval.

Can a discontinuous function be concave?

Concave functions can only be discontinuous at the endpoints of the defined interval.

Are all continuous functions convex?

due to general Convex functions are discontinuous They are also not necessarily continuous when defined on open sets of topological vector spaces. …but every convex function over the real numbers is lower semicontinuous over the relative interior of its valid domain, which in this case is equal to the domain of definition.

Are concave functions always continuous?

An alternative proof for this concave function is continuous within the relative Its domain first shows that it is a bounded small open set, then from boundedness and concavity, continuities are derived. …if f : C → R is concave, C ⊂ Rl is convex, and the interior is non-empty, then f is continuous on int(C).

Can a piecewise function be concave?

An important function is neither concave Convexity also frequently occurs in production and inventory models. This function is referred to here as a piecewise concave function, which can be considered as a generalization of the concave function. … This paper explores various properties of piecewise concave functions.

Continuous, discontinuous and piecewise functions

42 related questions found

Can a piecewise function be convex?

Any convex segment –Linear functions are convex.

How do you prove concave?

To judge whether 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.

What is a concave curve?

concave describe an inward curve; its opposite is convex, describing a curve that bulges outwards. They are used to describe soft, subtle curves, such as those in mirrors or lenses. …If you were to describe a bowl, you might say that the concave side has a large blue dot in the center.

Is the bump up or down?

(Video) Concavity, Knee, and Second Derivatives

A function is sunken (or convex) if it curves upwards. A function is concave (or just concave) if it curves down.

How do you prove convex?

Theorem 1. A function f : Rn → R is convex if and only if the function g : R → R given by g

What is a convex set with examples?

Equivalently, a convex set or convex region is the subset that intersects each line into a single (possibly empty) line segment. E.g, a solid cube is a convex set, but anything that is hollow or indented, such as a crescent, is not convex.

What is an example of a convex surface?

A convex shape is a shape in which all of its parts are « out ». In other words, it doesn’t have any part pointing inside. E.g, a complete pizza is a convex shape because its full contour (perimeter) points outward.

Can derivatives be discontinuous?

A basic example of a differentiable function with discontinuous derivatives is f(x)={x2sin(1/x) if x≠00 if x=0. The differentiation rule states that the function is differentiable away from the origin, the difference quotient can be used to show that it is differentiable at the origin, with the value f'(0)=0.

What does it mean if the derivative is not continuous?

The derivative of a function at a given point is the slope of the tangent at that point. So, if you can’t draw tangents, there are no derivatives – this happens in cases 1 and 2 below. … removable discontinuity – that’s a fancy term for a hole – like the holes in the functions r and s in the diagram above.

What is the derivative of a discontinuous function?

it is discontinuous x=0 (the limit limx→0f(x) doesn’t exist, so it’s not equal to f(0)), but if I find the derivative using the limit above, I get the left and right limits equal to 1. Therefore, the derivative exists.

Is concave a negative number?

concave down, because negative over the given interval. is concave upward because it is positive over the given interval.

What is concave up or down?

Concavity is related to the rate of change of the derivative of a function. The function f is concave upward (or upward) as the derivative f’ increases. … Again, f is concave down (or down) where the derivative f’ decreases (or equivalently, f »f, start superscript, prime, prime, end superscript negative).

What is concave down?

A function is concave if its graph is below its tangent. If it is important to know where the graph is concave up/down, so is where the graph changes from one to the other. This leads us to a definition. Definition: Inflection point.

What does a concave curve look like?

concave description an inward curved shape, like an hourglass. Convex describes a shape that curves outward, like a football (or rugby).

What is a convex curve?

parabolaa simple convex curve example.

Is the bowl concave or convex?

one with concave Bend inward, such as a spoon or bowl. The middle is thinner than the edges. Objects with a convex shape are objects that curve outward, such as a basketball or baseball.

What does a concave function look like?

A function of one variable is concave If each line segment connecting two points on its graph is not above the graph at any point. Symmetrically, a function of a single variable is convex if every line segment connecting two points on its graph lies below the graph at no point.

Is the function convex?

An intuitive definition: a function is called spaced convex If for all pairs of points on the graph, the line segment connecting the two points passes over the curve. curve. A convex function has an increasing first derivative, making it appear to be curved upwards.

What is a strictly concave function?

function is called strictly concave function if. for any and. For a function, the second definition simply states that for every point strictly between and , the point on the graph is above the line connecting the points and the sum.

How do you know if the derivative is continuous?

If all partial derivatives of a function exist in the neighborhood of point x0 and are continuous at point x0, then the function is differentiable at that point x0. Non-differentiable at (0, 0), but also all partial and directional derivatives exist.