1

Consider the function $g(x,u)$, where $g:\mathbb{R}^n \times U \mapsto \mathbb{R}$, $U\subset \mathbb{R}^m$ and is compact. $g$ is $C^2$ w.r.t $x$ and $u$. I am interested in deducing the properties of the function

\begin{equation} \tilde{g}(x) = \min_{u\in U} g(x,u) \end{equation}

More specifically: is $\tilde{g}$ continuous? The way I see it is that $u$ can be thought of as an index, and so you are trying to find the minimum of an uncountable number of continuous functions.

The minimum of two continuous functions is continuous. Is it correct to argue that given a countable number of continuous functions one can apply this argument to successive pairs of functions and deduce that the minimum of a countable number of continuous functions is also continuous?

What can one deduce about an uncountable number? Can anyone suggest a good textbook/reference?

MathsStudent
  • 189
  • I think trying to view your uncountable set as some sort of sequence is not going to work - even the minimum of countable many continuous functions may not be well-defined, and if it is, may not be continuous. However in your case $g$ should be continuous - try proving it directly, using the fact that its domain is a product. – Richard Rast Mar 11 '15 at 12:37
  • Note that (I believe) the $C^2$ condition is not needed. Your should only need compactness of $U$ and continuity of $g$. But it needs to be really continuous, not just continuous in each variable separately. – Richard Rast Mar 11 '15 at 12:38
  • What is the proof that the min of a countable number of continuous functions is not well-defined? Or an example? – MathsStudent Mar 11 '15 at 12:53
  • Think example. Say $f_n(x)=x^n$.The minimum is not defined on the negative reals, and it's defined but not continuous on the nonnegative reals (specifically discontinuous at one). – Richard Rast Mar 12 '15 at 03:27
  • 1
    In this thread the same question is solved for maximums instead. – andreshp Apr 20 '17 at 16:11
  • The minimum and maximum cases are equivalent because the minimum of $g(x, u)$ occurs at the same point as the maximum of $-g(x, u)$. – Paul Wintz Jul 27 '24 at 23:46

0 Answers0