The “epsilon-delta” nature of analysis can be daunting and unintuitive to students, as the heavy reliance on inequalities rather than equalities. But it occurred to me recently that one might be able to leverage the intuition one already has from “deals” – of the type one often sees advertised by corporations – to get at least some informal understanding of these concepts.

Take for instance the concept of an upper bound {X \leq A} or a lower bound {X \geq B} on some quantity {X}. From an economic perspective, one could think of the upper bound as an assertion that {X} can be “bought” for {A} units of currency, and the lower bound can similarly be viewed as an assertion that {X} can be “sold” for {B} units of currency. Thus for instance, a system of inequalities and equations like

\displaystyle 2 \leq Y \leq 5

\displaystyle X+Y \leq 7

\displaystyle X+Y+Z = 10

\displaystyle Y+Z \leq 6

could be viewed as analogous to a currency rate exchange board, of the type one sees for instance in airports:

Currency We buy at We sell at
{Y} {2} {5}
{X+Y} – {7}
{X+Y+Z} {10} {10}
{Y+Z} – {6}

Someone with an eye for spotting “deals” might now realize that one can actually buy {Y} for {3} units of currency rather than {5}, by purchasing one copy each of {X+Y} and {Y+Z} for {7+6=13} units of currency, then selling off {X+Y+Z} to recover {10} units of currency back. In more traditional mathematical language, one can improve the upper bound {Y \leq 5} to {Y \leq 3} by taking the appropriate linear combination of the inequalities {X+Y \leq 7}, {Y+Z \leq 6}, and {X+Y+Z=10}. More generally, this way of thinking is useful when faced with a linear programming situation (and of course linear programming is a key foundation for operations research), although this analogy begins to break down when one wants to use inequalities in a more non-linear fashion.

Asymptotic estimates such as {X = O(Y)} (also often written {X \lesssim Y} or {X \ll Y}) can be viewed as some sort of liquid market in which {Y} can be used to purchase {X}, though depending on market rates, one may need a large number of units of {Y} in order to buy a single unit of {X}. An asymptotic estimate like {X=o(Y)} represents an economic situation in which {Y} is so much more highly desired than {X} that, if one is a patient enough haggler, one can eventually convince someone to give up a unit of {X} for even just a tiny amount of {Y}.

When it comes to the basic analysis concepts of convergence and continuity, one can similarly view these concepts as various economic transactions involving the buying and selling of accuracy. One could for instance imagine the following hypothetical range of products in which one would need to spend more money to obtain higher accuracy to measure weight in grams:

Object Accuracy Price
Low-end kitchen scale {\pm 1} gram {\$ 5}
High-end bathroom scale {\pm 0.1} grams {\$ 15}
Low-end lab scale {\pm 0.01} grams {\$ 50}
High-end lab scale {\pm 0.001} grams {\$ 250}

The concept of convergence {x_n \rightarrow x} of a sequence {x_1,x_2,x_3,\dots} to a limit {x} could then be viewed as somewhat analogous to a rewards program, of the type offered for instance by airlines, in which various tiers of perks are offered when one hits a certain level of “currency” (e.g., frequent flyer miles). For instance, the convergence of the sequence {x_n := 2 + \frac{1}{\sqrt{n}}} to its limit {x := 2} offers the following accuracy “perks” depending on one’s level {n} in the sequence:

Status Accuracy benefit Eligibility
Basic status {|x_n - x| \leq 1} {n \geq 1}
Bronze status {|x_n - x| \leq 0.1} {n \geq 10^2}
Silver status {|x_n - x| \leq 0.01} {n \geq 10^4}
Gold status {|x_n - x| \leq 0.001} {n \geq 10^6}
{\dots} {\dots} {\dots}

With this conceptual model, convergence means that any status level of accuracy can be unlocked if one’s number {n} of “points earned” is high enough.

In a similar vein, continuity becomes analogous to a conversion program, in which accuracy benefits from one company can be traded in for new accuracy benefits in another company. For instance, the continuity of the function {f(x) = 2 + \sqrt{x}} at the point {x_0=0} can be viewed in terms of the following conversion chart:

Accuracy benefit of {x} to trade in Accuracy benefit of {f(x)} obtained
{|x - x_0| \leq 1} {|f(x) - f(x_0)| \leq 1}
{|x - x_0| \leq 0.01} {|f(x) - f(x_0)| \leq 0.1}
{|x - x_0| \leq 0.0001} {|f(x) - f(x_0)| \leq 0.01}
{\dots} {\dots}

Again, the point is that one can purchase any desired level of accuracy of {f(x)} provided one trades in a suitably high level of accuracy of {x}.

At present, the above conversion chart is only available at the single location {x_0}. The concept of uniform continuity can then be viewed as an advertising copy that “offer prices are valid in all store locations”. In a similar vein, the concept of equicontinuity for a class {{\mathcal F}} of functions is a guarantee that “offer applies to all functions {f} in the class {{\mathcal F}}, without any price discrimination. The combined notion of uniform equicontinuity is then of course the claim that the offer is valid in all locations and for all functions.

In a similar vein, differentiability can be viewed as a deal in which one can trade in accuracy of the input for approximately linear behavior of the output; to oversimplify slightly, smoothness can similarly be viewed as a deal in which one trades in accuracy of the input for high-accuracy polynomial approximability of the output. Measurability of a set or function can be viewed as a deal in which one trades in a level of resolution for an accurate approximation of that set or function at the given resolution. And so forth.

Perhaps readers can propose some other examples of mathematical concepts being re-interpreted as some sort of economic transaction?