Question of double limits I post a problem concerning a possible generalization of the question of interchanging in double limits. Given a sequence of functions $\{f_{j}\}_{j}$ on an interval $I$ and a point $a\in I$, is it true that
$$\liminf_{j→∞}(\liminf_{x→a}f_{j}(x))≤\liminf_{x→a}(\liminf_{j→∞}f_{j}(x))≤\limsup_{x→a}(\liminf_{j→∞}f_{j}(x))≤\limsup_{j→∞}(\limsup_{x→a}f_{j}(x))?$$
Any reference would be helpful. Thank you very much.
 A: The clue suggesting that this should not be true is that why would you except the situation in variables $x$ and $j$ be asymmetric; they have more-or-less the same role.

Let us look at double sequences first. Is it true that 
$$\limsup\limits_{i\to\infty} \limsup\limits_{j\to\infty} x_{i,j} = 
\limsup\limits_{j\to\infty} \limsup\limits_{i\to\infty} x_{i,j}?$$
Certainly not. An easy counterexample is taking
$$x_{i,j}=
  \begin{cases}
    1 & \text{if }i\le j, \\
    0 & \text{if }i>j.
  \end{cases}
$$
Then we get $\limsup\limits_{j\to\infty} \limsup\limits_{i\to\infty} x_{i,j}=0$ and $\limsup\limits_{i\to\infty} \limsup\limits_{j\to\infty} x_{i,j}=1$.

Are we able modify the above example to get
$$\limsup\limits_{j\to\infty} \limsup\limits_{x\to a} f_j(x)=0\text{ and }\limsup\limits_{x\to a} \limsup\limits_{j\to\infty} f_j(x)=1?$$
We can simply define the function by
$$f_j(x)= x_{ij} \text{ for }x\in (a+1/i+1,a+1/i].$$
If you wish to have values also on the left from $a$, you can modify it symmetrically.
