Given a discontinuos function, show that it is not concave

  • #1
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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 [itex]\alpha, x_{1}, x_{2}[/itex] such that

[itex]\alpha f(x_{1})+(1-\alpha)f(x_{2}) \geq f(\alpha x_{1}+(1-\alpha)x_{2}) [/itex]

Now, let [itex]x_{1}[/itex] be a point of discontinuity of f. Thus

[itex]lim_{x \rightarrow x_{1}}f(x) \neq f(x_{1}) [/itex]

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

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

[itex]\alpha f(x_{1})+(1-\alpha)f(x_{2}) \geq f(\alpha x_{1}+(1-\alpha)x_{2}) [/itex].

Is this correct? Can you provide any hints?

Thank you

A
 
Last edited:

Answers and Replies

  • #2
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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
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Hello,

It's a class in Microeconomic Theory

Thanks
 
  • #4
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so what you are trying to show is that if the function is not continuous, that it is convex?
 
  • #5
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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
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at this level of mathematics.. if you can even understand the question, you should get an "A"..

:)
 
  • #7
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what text are you studying from?..
 
  • #8
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Its called Microeconomic Theory by Mas-Collel, Winston, and Green.

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

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