Can a discontinuous function be concave or convex?

Can a convex function be discontinuous? – Quora. Yes, on the boundary of its domain. Consider: and that is because convexity of a function is a property of its epigraph (the points above the graph of the function).

Is convex function always 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.

Can concave functions be discontinuous?

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

Is a concave function always 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 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.

Can a piecewise function be concave?

An important function which is neither concave nor convex often arises in production and inventory models. This function is herein called piecewise concave and can be considered to be a generalization of the concave function. Various properties of piecewise concave functions are explored in this paper.

Can a function be concave and convex?

Absolutely ! Take a look at a function that is both convex and concave on . A simple example of such a function is given by all the linear functions : is a perfectly fit example of a function that is both convex and concave.

Are all convex functions Lipschitz?

Every convex function f defined on an open convex set in Rn is locally Lipschitz. A different recipe yields the same result with less work and applies in much more general spaces.

Is a convex function Lipschitz?

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. Hence x2 + |x| is strongly convex.

Can a piecewise function be convex?

Yes, it can be convex.