Can the supremum of continuous functions be discontinuous at every point of an interval?
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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?
real-analysis
asked Aug 31 at 23:45
Angelo
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up vote
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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?
real-analysis
asked Aug 31 at 23:45
Angelo
312
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
1
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
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
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?
real-analysis
asked Aug 31 at 23:45
Angelo
312
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?
real-analysis
real-analysis
asked Aug 31 at 23:45
Angelo
312
asked Aug 31 at 23:45
Angelo
312
edited Sep 1 at 0:13
asked Aug 31 at 23:45
Angelo
312
asked Aug 31 at 23:45
Angelo
312
asked Aug 31 at 23:45
Angelo
312
312
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
1
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
add a comment |Â
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
1
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
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
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
1
1
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
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, 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
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
@StanleyYaoXiao The OP wants the function to be discontinuous at every point in the interval.
– Noah Schweber
Aug 31 at 23:58
add a comment |Â
1 Answer
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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
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
1
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
add a comment |Â
1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
23
down vote
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
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
1
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
add a comment |Â
up vote
23
down vote
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
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
1
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
add a comment |Â
up vote
23
down vote
up vote
23
down vote
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
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
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
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
edited Sep 1 at 7:11
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
answered Sep 1 at 0:31
Martin Sleziak
2,74432028
2,74432028
1
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
add a comment |Â
1
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
1
1
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
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
add a comment |Â
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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
1
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