# Discontinuous Almost-Everywhere/ Unbounded in $L^{1}(\mathbb{R})$

Let $L^{1}(\mathbb{R})$ be defined as usual, with the equivalence relation : $f \approx g$ if and only if $f(x) = g(x)$ almost everywhere.

Is there a class in $L^{1}(\mathbb{R})$ such that every element is discontinuous almost everywhere ? Is there another class such that the every element is discontinuous everywhere ?

Thank you very much :)

EDIT : Now I know about the existence of a set $A$ that is dense, has measure greater than zero and its dense complement also has measure greater than zero. It solves this problem. (thanks to David Mitra)

EDIT 2 : Can you help me now with another question ? Is there a class in $L^1(\mathbb{R})$ such that every element is unbounded in every non-empty open interval in $\mathbb{R}$ ?

• What is a representant? – Chris Janjigian May 6 '13 at 16:44
• A member that defines a partition in the equivalence relation – thetruth May 6 '13 at 16:46