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An open set in ##\mathbb{R}## is any set which can be written as the union of open intervals ##(a,b)## with ##a<b##.
A subset of ##\mathbb{R}## is called a ##G_\delta## set if it is the countable intersection of open sets.
Prove that if a set ##A\subseteq \mathbb{R}## is a ##G_\delta## set then there exists a function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##f## is continuous at all points of ##A## and discontinuous at all points of ##\mathbb{R}\setminus A##.
The converse holds as well and has a very easy proof using the Baire Category theorem. This characterizes the continuity set of a function.
This is a classis analysis result, so it can be easily googled if you want to. But I trust you to play it fair
A subset of ##\mathbb{R}## is called a ##G_\delta## set if it is the countable intersection of open sets.
Prove that if a set ##A\subseteq \mathbb{R}## is a ##G_\delta## set then there exists a function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##f## is continuous at all points of ##A## and discontinuous at all points of ##\mathbb{R}\setminus A##.
The converse holds as well and has a very easy proof using the Baire Category theorem. This characterizes the continuity set of a function.
This is a classis analysis result, so it can be easily googled if you want to. But I trust you to play it fair