For motivation, imagine a pair of analog dials that turn continuously from 0 to 1 - how can you turn those dials in such a search pattern, that eventually every pair of possible dial positions is visited?
Let us define $T$ as the set of real numbers greater or equal to 0.
Let us define $U$ as the set of real numbers between 0 and 1, inclusive.
Let's say that a function, $f(x)$, is pretty if all of:
- is a unary function
- has a domain of $T$
- has a range of $U$
- is continuous
- is differentiable.
For example, $(sin(x)+1)\over 2$, is pretty.
Does there exist a pair of pretty functions, $p$ and $q$, such that for any two elements of $U$, $a$ and $b$, there exists some element of $T$, t, such that both $p(t) = a$ and $q(t) = b$ ?
If yes, what is one such pair of functions?
if not, whats a sketch of a proof?