A challenge for those who have seen a bit of analysis before.

A map \(f\colon X\to Y\) is continuous (everywhere) if

\[ \forall x\in X\, \forall \epsilon>0\,  \exists \delta>0\,  \forall x’\in X\,  d(x,x’)<\delta \Longrightarrow d(f(x),f(x’))<\epsilon. \]

On the other hand, it is uniformly continuous if

\[\forall \epsilon>0\,  \exists \delta>0\, \forall x\in X\, \forall x’\in X\,  d(x,x’)<\delta \Longrightarrow d(f(x),f(x’))<\epsilon. \]

Check your understanding of quantifier-ology by convincing yourself that uniformly continuous implies continuous, but not conversely (e.g. the map \(x\mapsto x^2\) from the real line to itself is continuous but not uniformly continuous).

The argument we gave for the continuity of the Peano curve definitely does not prove uniform continuity, because (as we said in class), for fixed \(x\),  the number \(\delta\) such that if \(|x-x’|<\delta\) then \(x\) and \(x’\) agree up to \(m\) digits in their ternary expansion depends on the ternary expansion of \(x\).  On the other hand, it is a general theorem that when \(X\) is compact, e.g. the unit interval, every continuous map actually is uniformly continuous.  Explain this apparent inconsistency.

Here’s a write-up of the story I told at the end of class (though I seem to have relocated it from Russia to Wales….)

In a certain country there are two cities — call them Aberystwyth and Betws-y-Coed — and two roads that join them: the “low road” and the “high road”.

In A  dwell two lovers, Maelon and Dwynwen, who must  travel to B: M  by the high road, and D by the low.  So great is the force of their love that if at any instant they are separated by ten miles or more, they will surely die.

As well as a pair of lovers, our story contains a pair of sworn enemies, Llewelyn and John.  As our story begins, L is in A, J is in B, and they must exchange places, L traveling from A to B via the high road while J travels from B to A via the low road.   So great is the force of their hatred that if at any instant they are separated by ten miles or less, they will surely die.

Prove that tragedy is inevitable.  At least two people will end up dead.

We’ll talk about the solution at the beginning of the review session on Tuesday (tomorrow).  To get your thinking in gear, try to answer the following question: Maelon and Dwynwen consult a map, and they discover that every point of the high road is within 10 miles of some point on the low road, and every point on the low road is within 10 miles of some point on the high road.  This is an obvious necessary condition for their safety: is it also sufficient? In other words, given this information, is there a strategy that they can follow to guarantee them a safe journey? If yes, try to prove that your strategy always works; if no, find an example where the condition is satisfied but M and D are still doomed.

PS: For more about true love and implacable hatred, I recommend The Princess Bride.