Skip to main content
MathOverflow

Return to Question

Post Closed as "Not suitable for this site" by Igor Rivin, Stefan Kohl♦, Alexandre Eremenko, Gro-Tsen, Andrés E. Caicedo
replaced "functional analysis" tag
Link
edit approved Oct 5, 2017 at 9:56
Jules Lamers
edit approved Oct 5, 2017 at 9:56
Jules Lamers
  • 2k
  • 1
  • 18
  • 35
edited body
Source Link
Zetapology
Zetapology
  • 675
  • 4
  • 13

By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(x,y)$. This is merely so that the function does not have to be differentiable.

If it's not possible to have a function be continuous but not convex or concave on any infinitenonempty interval, is it possible to construct a function which is not convex or concave on any infinitenonempty interval? Has one already been created?

If it is possible to have a function be continuous but not convex or concave on any infinitenonempty interval, is it possible to construct it? Has one already been created?

Is it possible to have one of these functions be differentiable once?

By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(x,y)$. This is merely so that the function does not have to be differentiable.

If it's not possible to have a function be continuous but not convex or concave on any infinite interval, is it possible to construct a function which is not convex or concave on any infinite interval? Has one already been created?

If it is possible to have a function be continuous but not convex or concave on any infinite interval, is it possible to construct it? Has one already been created?

Is it possible to have one of these functions be differentiable once?

By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(x,y)$. This is merely so that the function does not have to be differentiable.

If it's not possible to have a function be continuous but not convex or concave on any nonempty interval, is it possible to construct a function which is not convex or concave on any nonempty interval? Has one already been created?

If it is possible to have a function be continuous but not convex or concave on any nonempty interval, is it possible to construct it? Has one already been created?

Is it possible to have one of these functions be differentiable once?

Source Link
Zetapology
Zetapology
  • 675
  • 4
  • 13

Are there functions which are neither convex nor concave everywhere but are continuous?

By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(x,y)$. This is merely so that the function does not have to be differentiable.

If it's not possible to have a function be continuous but not convex or concave on any infinite interval, is it possible to construct a function which is not convex or concave on any infinite interval? Has one already been created?

If it is possible to have a function be continuous but not convex or concave on any infinite interval, is it possible to construct it? Has one already been created?

Is it possible to have one of these functions be differentiable once?