0
$\begingroup$

$$f(x)=-\sum_{i=1}^{n}log(b_i-a_i^Tx)$$ $dom\ f_0={Ax

How to prove that the sublevel sets of $f_0$ is closed?

Generally, for a given function, how to show its sublevel sets are closed or not?

  • 0
    Do you mean closed relative to the domain or closed relative to $\mathbb{R}^n$? Typically one appeals to continuity to show closure. – 2017-02-20
  • 0
    @copper.hat are you saying a continuous function always has its sublevel sets closed? – 2017-02-20
  • 1
    The sublevel sets are of the form $f^{-1} ((-\infty, \alpha])$ and since $(-\infty, \alpha]$ is closed, it follows that the sub level sets are closed if $f$ is continuous. Closed means closed relative to the domain in this case. – 2017-02-20

0 Answers 0