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Supposed to be simple but seems a bit tricky

  1. May 7, 2010 #1
    Let [tex]f:{R^n} \to R[/tex] be a continuous function on some [tex]D \subset {R^n}[/tex].
    Now, define
    [tex]epi\left( f \right) = \left\{ {\left( {x,r} \right) \in {R^{n + 1}}:x \in {R^n},r \ge f\left( x \right)} \right\}[/tex]​
    I want to prove to myself (not homework) that if [tex]D[/tex] has a non-empty interior (that is, there exist interior points in [tex]D[/tex]), so also [tex]epi\left( f \right)[/tex] has a non-empty interior.
    I think it is supposed to be easy but I get stuck.

    I started with the following simple argument. If [tex]D[/tex] has a non-empty interior, then there exist [tex]{x_0} \in D[/tex] and [tex]r > 0[/tex] in [tex]R[/tex] such that [tex]B\left( {{x_0},r} \right) \subset D[/tex] (that is, the ball with radius [tex]r[/tex] around [tex]{x_0}[/tex] is contained in [tex]D[/tex]). Now, because [tex]f[/tex] is continuous, there also exist [tex]0 < {r_2} < r[/tex] in [tex]R[/tex] such that [tex]f\left( x \right) < f\left( {{x_0}} \right) + 1[/tex] for all [tex]x \in B\left( {{x_0},{r_2}} \right)[/tex].
    Now, I want to show that [tex]\left( {{x_0},f\left( {{x_0}} \right) + 1} \right) \in epi\left( f \right)[/tex] but I do not have a good idea how to continue.

    Thanks a lot in advance! :smile:

    p.s. If you think that it is more appropriate to put this question in homework section, please notify me.
     
  2. jcsd
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