limes inferior & limes superior

Question

If $U=[-1,1]$ and

$J(u)=u, u\in (0,1]$
$J(u)=a, u=0$
$J(u)=1-u, u\in [-1,0)$

how to calculate limes inferior and limes superior of $J(u)$?

Is this correct:

I choose arbitrary sequence $\{u_k\}\in U$ such that $\displaystyle \lim_{k \to +\infty} u_k=0$

$\displaystyle \lim_{k \to +\infty} J(u_k)=\displaystyle \lim_{k \to +\infty} u_k=0$ for $u_k\in (0,1]$

$\displaystyle \lim_{k \to +\infty} J(u_k)=\displaystyle \lim_{k \to +\infty} (1-u_k)=1$ for $u_k\in [-1,0)$

$\lim\inf J(u_k)=0$
$\lim\sup J(u_k)=1$

Is this ok?

I need these limits to show semi-continuity of a function $J(u)$ in $u=0$.
I have to show that $J$ is lower semi-continuous for $a\leq 0$ and upper semi-continuous for $a\geq 0$.
If I do that as I wrote above, is it correct?

Answer 1

Your work is right, but the problem statement must be wrong, because if $a$ is between 0 and 1, the function is neither upper nor semi continuous. If $a$ is greater than 1, though, it is in fact upper-semi continuous.