$u_n\to f\in L^1([0,1])$ but $u_n'\to g\notin L^1([0,1])$?

Question

Is it possible to have a sequence of functions $(u_n)_{n\ge 1} \subset C^1([0,1])$ such that $u_n\to f\in L^1([0,1])$ but the sequence of derivatives are such that $u_n'\to g\notin L^1([0,1])$?

Answer 1

Yes, for example $u_n(x)=(1+\frac{1}{n})e^{-1/x}$ if $0<x$ and $u_n(0)=0$.