Prove that the limit function is discontinuous

Question

Consider the space $C([-1,1])$ with metric $d(x,y)=\int_{-1}^1|x(t)-y(t)|dt$. I need to show it's not complete. I proved that the following sequence is Cauchy: $x_n(t)=-1$ for $t\in [-1,-1/n]$, $x_n(t)=nt$ for $t\in [-1/n,1/n]$, $x_n(t)=1$ for $t\in [1/n,1]$. I guess it's limit is discontinuous. But how do I prove this?

Answer 1

Suppose for sake of contradiction that $x \in C([-1,1])$ is such that $d(x_n, x) \to 0$. Show $x(t)=-1$ for $t \in (-1, 0)$ and $x(t) = 1$ for $t \in (0,1)$.