# Given a discontinuos function, show that it is not concave

1. Oct 4, 2011

### michonamona

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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

Last edited: Oct 4, 2011
2. Oct 4, 2011

### 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..

3. Oct 4, 2011

### michonamona

Hello,

It's a class in Microeconomic Theory

Thanks

4. Oct 4, 2011

### Hammie

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

5. Oct 4, 2011

### 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')

6. Oct 4, 2011

### Hammie

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

:)

7. Oct 4, 2011

### Hammie

what text are you studying from?..

8. Oct 4, 2011

### 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.