Let $I$ be the unit interval. I developed a method to simplify problems of the form "Construct a function $f:I \rightarrow I$ satisfying given properties". These properties should only be expressible in the structure $(I,\leq)$, i.e.: not using any properties of $I$ beyond its structure as a dense total ordering. Finding examples of such problems will help me test out my method, so I can iron out any bugs and make it more efficient.
A couple of examples of such a problem would be the trivial "Construct a function that is constant on $I$." to the less trivial "Construct a function $f:I\rightarrow I$ that maps every open interval to the whole of $I$". The latter example satisfies the requirements because the condition can be expressed as "$\forall a \forall b \forall y [(a<b) \implies \exists x (a<x<b \wedge f(x)=y)]$".
Any interesting examples of such problems would be appreciated, and I would be especially happy with a wide range of difficulties in your examples.
Thank you !!