I am trying to prove that if $u:(0,1)\to\mathbb{R}$ lies in $W^{1,1}(0,1)$, then $u\in C(0,1)$. Is there any help anybody can offer?


Thanks.



Every function in W^{1,1}(0,1) is continuous on (0,1)
Since $u'\in L^1(0,1)$, you find from the Lebesgue differentiation theorem that
$$
\int\_{1/2}^x u'(t) dt=u(x)+Cst,\quad x\in(0,1).
$$
As a result $u$ is a continuous function and the constant above is $-u(1/2).$


