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Homework Help: Given a discontinuos function, show that it is not concave

  1. Oct 4, 2011 #1
    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 [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

    Last edited: Oct 4, 2011
  2. jcsd
  3. Oct 4, 2011 #2
    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..
  4. Oct 4, 2011 #3

    It's a class in Microeconomic Theory

  5. Oct 4, 2011 #4
    so what you are trying to show is that if the function is not continuous, that it is convex?
  6. Oct 4, 2011 #5
    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')
  7. Oct 4, 2011 #6
    at this level of mathematics.. if you can even understand the question, you should get an "A"..

  8. Oct 4, 2011 #7
    what text are you studying from?..
  9. Oct 4, 2011 #8
    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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