# 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 $f_n:\mathbb{R}\to\mathbb{R}$ such that $f(x) = \sup_{n \in \mathbb{N}} f_n(x)$ is a discontinuous function at every point of a subinterval of $\mathbb{R}$ ?

If such a sequence does not exist, how is it possible to prove it?

• Are you asking for a sequence of continuous real valued functions $\{f_n\}$ such that $f(x) = \sup_{n \in \mathbb{N}} f_n(x)$ is discontinuous on a subinterval of $\mathbb{R}$? – Stanley Yao Xiao Aug 31 '18 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 '18 at 23:51
• Yes, I am asking for a sequence of continuous real valued functions such that $f(x) = \sup_{n \in \mathbb{N}} f_n(x)$ is discontinuous on a subinterval of $\mathbb{R}$ – Angelo Aug 31 '18 at 23:54
• Yes, Martin, I mean the answer to that question. – Angelo Aug 31 '18 at 23:55
• @StanleyYaoXiao The OP wants the function to be discontinuous at every point in the interval. – Noah Schweber Aug 31 '18 at 23:58

## 1 Answer

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 $f\colon \mathbb R\to\mathbb R$ 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 $g_n = \sup\{f_1,\dots,f_n\}$ that $f=\lim g_n$ pointwise. – Dirk Werner Sep 4 '18 at 18:54