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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

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