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Martin Sleziak
Martin Sleziak
06:18
For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website
 
8 hours later…
user193319
user193319
14:19
Problem:Show that there cannot be a sequence of continuous functions $f_n : [0,1] \to \Bbb{R}$ such that $\{f_n(x)\}$ is decreasing for every $x \in [0,1]$, $\lim_{n \to \infty} f_n(x) = 1$ if $x \in [0,1[ \cap \Bbb{Q}$, and $\lim_{n \to \infty} f_n(x) = 0$ if $x \in [0,1] \setminus \Bbb{Q}$.
I could use a hint.
Martin Sleziak
Martin Sleziak
14:30
@user193319 I'd guess Dini's theorem could help here.
I see that in the main chatroom you got advice based on functions in the first Baire class (which does not use monotonicity).
in Mathematics, 2 mins ago, by Alessandro Codenotti
functions which are pointwise limit of continuous functions are called Baire one functions, it's not trivial to show but it's well known that if $f$ is Baire one then every closed interval contains a point of continuity of $f$, so in particular it cannot be everywhere discontinuous
in Mathematics, 1 min ago, by Alessandro Codenotti
functions which are poinwise limit of Baire one functions are called Baire two (and so on), the indicator function of the rationals that you're looking at is Baire two for example
user193319
user193319
Ah, that's a nice theorem!
Martin Sleziak
Martin Sleziak
WP: Baire function
in Mathematics, 2 mins ago, by Alessandro Codenotti
I found a nice reference
Johnny Hu: Baire One Functions; whitman.edu/Documents/Academics/Mathematics/huh.pdf

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