Wrapped Func Eqn

Wrapped Functional Equations is a unit on a specific (but very common) type of FEs.

Content

Evan defines a functional equation to be “fully wrapped” if there are no “free variables” (in the sense that all terms are wrapped by the function in some way). This unit aims to develop techniques to tackle such functional equations.

Over rationals and integers, they generally admit inductive-style solutions. However, the unit’s job is to unwrap the mystery of real domains.

In particular, functional equations over positive reals tend to use more involved techniques than those over all reals, so you’ll be dealing with a lot of them.

Some types of arguments you’ll be using include:

• Inequalities (monotony or boundedness);

• Additivity (the classical “reducing to Cauchy”);

• Careful injectivity (you’ll notice that claiming injectivity often allows for an easy finish; what’s hard is proving it).

Notable Problems

Some poster examples include:

• IMO 2017/2: beautiful but surprisingly difficult, this Walkthrough FE bears historical significance;

• USAMO 2018/2: over positive reals and very technical, this Walkthrough FE will test your patience;

• RMM 2019/5: an excellent exercise of one of the highlighted techniques, this is a Required Problem;

• Shortlist 2003 A5: with a set of surprising conditions, this FE takes you on a fun ride.

Recommendations

This is an average difficulty (but essential) D-level unit. It is harder than its unwrapped counterpart (of B and D-levels) but easier than its pathological cousin(s).

If you haven’t done either already (and especially if you haven’t done FEs before), you should start with the B or D version of the classical unit out of the FEs unit batch. Afterwards, follow up with this one (which can itself be followed up by Monsters).

All in all, unlocking this unit is recommended if you’re fairly comfortable with standard FEs. The techniques showcased here are extremely important, surpassing the basics but crucial nonetheless.