Given a discontinuos function, show that it is not concave

[michonamona]

Homework Statement

Let f be a function from (1,0) to (1,0). Suppose that f is discontinuous. Show that f is not concave.

Homework Equations

The Attempt at a Solution

Let f:(0,1)-->(0,1). Suppose f is discontinous. Show that it is not concave.I've been working on this problem for over an hour. This is what I got so far.

What I want to show is the following:

There exists \alpha, x_{1}, x_{2} such that

\alpha f(x_{1})+(1-\alpha)f(x_{2}) \geq f(\alpha x_{1}+(1-\alpha)x_{2})

Now, let x_{1} be a point of discontinuity of f. Thus

lim_{x \rightarrow x_{1}}f(x) \neq f(x_{1})

What I'm trying to show is that we can take an epsilon-neighborhood about f(x_{1}), call it N_{\epsilon}(f(x_{1})), small enough so that for a given \alpha, such that

f(\alpha x_{1}+(1-\alpha)x_{2}) \in N_{\epsilon}(f(x_{1})), then

\alpha f(x_{1})+(1-\alpha)f(x_{2}) \geq f(\alpha x_{1}+(1-\alpha)x_{2}).

Is this correct? Can you provide any hints?

Thank you

A

 

[Hammie]

what kind of class is this problem from? A little context might help know where to go. I might be able to help you with showing a midpoint convex function is convex if it is continuous..

 

[michonamona]

Hello,

It's a class in Microeconomic Theory

Thanks

 

[Hammie]

so what you are trying to show is that if the function is not continuous, that it is convex?

 

[michonamona]

I'm trying to show that if the function f is discontinuous, then it cannot be concave.

Here's the general definition of concavity

Let f be a function of many variables defined on the convex set S. Then f is
concave on the set S if for all x ∈ S, all x' ∈ S, and all λ ∈ (0,1) we have

f ((1−λ)x + λx') ≥ (1−λ) f (x) + λ f (x')

 

[Hammie]

at this level of mathematics.. if you can even understand the question, you should get an "A"..

:)

 

[Hammie]

what text are you studying from?..

 

[michonamona]

Its called Microeconomic Theory by Mas-Collel, Winston, and Green.

I thought concavity was a general mathematical property, I learned it in Real Analysis.