Can the supremum of continuous functions be discontinuous at every point of an interval?
Can the supremum of continuous functions be discontinuous at every point of an interval?
Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued functions $f_n:mathbbRtomathbbR$ such that $f(x) = sup_n in mathbbN f_n(x)$ is a discontinuous function at every point of a subinterval of $mathbbR$ ?
If such a sequence does not exist, how is it possible to prove it?
When you mention Peter Luthy's example, you mean the answer to this question, right: Can the supremum of continuous functions be discontinuous on a set of positive measure?
– Martin Sleziak
Aug 31 at 23:51
Yes, I am asking for a sequence of continuous real valued functions such that $f(x) = sup_n in mathbbN f_n(x)$ is discontinuous on a subinterval of $mathbbR$
– Angelo
Aug 31 at 23:54
Yes, Martin, I mean the answer to that question.
– Angelo
Aug 31 at 23:55
@StanleyYaoXiao The OP wants the function to be discontinuous at every point in the interval.
– Noah Schweber
Aug 31 at 23:58
1 Answer
1
Since the function $f$ is supremum of a set of continuous functions, it is lower-semicontinuous.1
Every lower semicontinuous function belongs to the first Baire class.2
If $fcolon mathbb Rtomathbb R$ is of the first Baire class, then the set $D_f$ of the points of discontinuity is a meager set.3
In particular, $D_f$ cannot be an interval.
1Theorem 10.3 in van Rooij-Schikhof: A Second Course on Real Functions. Mathematics Stack Exchange: To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous or Show that the supremum of a collection of lower semicontinuous function is lower semicontinuous.
2Theorem 10.6 and Exercise 11.E in van Rooij-Schikhof; Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions on Mathematics Stack Exchange
3Theorem 11.4 in van Rooij-Schikhof; MathOverflow: Points of continuity of Baire class one functions
One can bypass the ``lsc implies Baire-1'' step by noting directly that Angelo's $f$ is Baire-1, since one has for $g_n = supf_1,dots,f_n$ that $f=lim g_n$ pointwise.
– Dirk Werner
Sep 4 at 18:54
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Are you asking for a sequence of continuous real valued functions $f_n$ such that $f(x) = sup_n in mathbbN f_n(x)$ is discontinuous on a subinterval of $mathbbR$?
– Stanley Yao Xiao
Aug 31 at 23:51