Prove/disprove statements regarding continuous functions.

Prove/disprove statements regarding continuous functions.

I found some old math tests from my school years and thought it might be
fun to see what I still remember. The answer is simply, not as much as I
hoped for. I'm having trouble proving/disproving these statements, not
sure where to go with it.
If f is a continuous function on the closed interval [a,b] then f([a,b] =
[f(a),f(b)] or f([a,b]) = [f(b),f(a)].
There exists a function f that maps ]0,1[ to R/Z+
Is there a continuous function that maps ]0,10] to R?
I'm pretty sure that when I get 2, 3 will follow closely, hopefully at least.