0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 0 0 0
0 0 2 0 0
0 0 0 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 0 0 0
0 0 3 0 0
0 0 0 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 0 0 0
0 0 4 0 0
0 0 0 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 1 0 0
0 1 1 1 0
0 0 1 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 2 0 0
0 2 2 2 0
0 0 2 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 3 0 0
0 3 3 3 0
0 0 3 0 0
0 0 0 0 0
 
0 0 0 0 0
0 0 4 0 0
0 4 4 4 0
0 0 4 0 0
0 0 0 0 0
 
0 0 1 0 0
0 2 2 2 0
1 2 3 2 1
0 2 2 2 0
0 0 1 0 0

Could be any rule between the 8th and 9th step so hard to tell.

It could be, but it's not guaranteed to be.

Also, in a couple more steps there's a singularity with:

_ 4 _
4 4 4
_ 4 _

Where we have no obvious indication of how it would proceed.

Edit:

More accurately:

_ _ 4 _ _
_ 4 4 4 _
4 4 4 4 4
_ 4 4 4 _
_ _ 4 _ _

Errr wtf is this please educate me

0 0 4 0 0
0 4 4 4 0
4 4 4 4 4
0 4 4 4 0
0 0 4 0 0

|
V

0 2 2 2 0
2 3 5 3 2
2 5 5 5 2
2 3 5 3 2
0 2 2 2 0

The question is, how are 5's dealt with?

However this pattern will never be reached, so I'm not sure that it matters. OP's question is just whether it is continuous, which it is.

The next steps should be

0 0 2 0 0
0 3 3 3 0
2 3 4 3 2
0 3 3 3 0
0 0 2 0 0

0 0 3 0 0
0 4 5 4 0
3 5 1 5 3
0 4 5 4 0
0 0 3 0 0

Once again we have the questions of should we expand 5s straight away or leave them to the next turn. In the examples we have no situation where 5s are generated, so we don't know exactly how to proceed.

I expect this would be correct;

0 1 4 1 0
1 2 4 2 1
4 4 1 4 4
1 2 4 2 1
0 1 4 1 0

I didn't expect anyone to look so far into it. Props to you, Cow.

I was actually considering making a little program to simulate the spread and see how it would behave.

To clarify for anyone who's in the dark, I came up with this virus kind of expansion in which a number accumulates until it has enough capacity to spread to it's neighbors without exhausting itself. I suppose considering this that 5s would be distributed immediately, following the spread-without-exhaustion rule.

This is like blowing my mind, awesome stuff here.
Someone teach me D:

Hucota you should google "game of life"

I'm familiar with it. Pretty nifty simulation.