Engineering applications of epsilon-delta?

I would think that an engineer would commonly want to investigate a range of possible options: what $\delta$ would I need to get $|f(x)-f(x)|< \epsilon_1,\epsilon_2,$ or $\epsilon_3$? These might represent different specifications needed for different applications, or just part of a cost/benefit analysis. It is easier to solve one problem using a variable $\epsilon$ than to redo the same calculations for $3$ different particular values of $\epsilon$.

EDIT:

An engineer really wouldn't ever care about a bare existence theorem. They are only interested in constructive mathematics. The constructive way to prove that $\forall \epsilon >0 \, \exists \delta >0\, |x-c| < \delta \implies |f(x)-f(c)|<\epsilon$ is to produce a function $T:\mathbb{R}^+ \to \mathbb{R}^+$ for which, for all $\epsilon$, $|x-c|<T(\epsilon) \implies |f(x)-f(c)|<\epsilon$. In other words it is useless to know that some $\delta$ exists for a given $\epsilon$ unless you have a way to find it ($T$). Oftentimes there will be greater cost associated with a smaller $\delta$, and we will want to find the largest $\delta$ which will get the job done. In other words, it is desirable to find an explicit formula for $$T(\epsilon) = \sup \{\delta >0: |x - c|<\delta \implies |f(x) - f(c)|<\epsilon\}$$

if possible. Sometimes explicitly calculating this supremum is impossible and we have to settle for a suboptimal $T$.

A practical example:

We are designing a rocket to be launched with an initial velocity of exactly $1800 \, \frac{\textrm{m}}{\textrm{s}}$. We want to hit a target exactly $300 \, \textrm{km}$ away. To minimize civilian casualties we would like the blast radius of the payload to be as small as possible. However, the smaller the blast radius we specify the more precisely we will need to control the altitude ($\theta$). Engineering more precise control of that angle is costly, so we will need to conduct a cost/benefit analysis.

1. Assuming flat ground and no air resistance, what value $\theta$ would be needed to hit the target exactly?
2. We are interested in exploring a range of possible payload strengths. Call the blast radius of the payload $\epsilon$ meters. What is the maximum tolerable error ($\delta$ radians) of $\theta$ to make sure that our target is caught in the blast?

The best answer to this question is to conscientiously object to perform the calculation. The second best answer will reproduce a standard $\epsilon$-$\delta$ style proof of the continuity of the function $R(\theta) = \frac{v_0 \sin(2\theta)}{g}$ at the value of $\theta_0$ which solves $R(\theta_0) = 300$.

Note: There are some obvious objections to this particular problem on practical grounds:

1. We assumed exact knowledge of the initial velocity and distance to the target.
2. We assumed no air resistance and perfectly flat ground (no elevation change between rocket origin and target).
3. We implicitly assume that any blast radius $\epsilon$ meters would be practically relevant, with smaller $\epsilon$ having greater benefit in terms of civilian safety. There is, of course, some lower limit to what is practical here. An $\epsilon$ of $0.01$ is no safer than an $\epsilon$ of $0.1$. In reality we probably only need to worry about $\epsilon \in [0.1, 50]$.

Even so, I hope you can see that the basic form of the $\epsilon$-$\delta$ argument does mirror the sort of analysis that a practical user of mathematics would need to employ when they are thinking about error in their measurements. Making this connection is, in my opinion, important to motivate the $\epsilon$-$\delta$ definition of continuity. In fact, we could give the following alternative definition of the continuity of a function $f$ at a point $x = x_0$ which aligns with this motivation:

Definition: A function $f:(a,b) \to \mathbb{R}$ is continuous at $x_0 \in (a,b)$ if and only if there is a function $T:\mathbb{R}^+ \to \mathbb{R}^+$ so that, for all $\epsilon > 0$ if $|x - x_0| < T(\epsilon)$ then $|f(x) - f(x_0)| < \epsilon$.

In other words, if you want to control $f(x)$ to within $\pm \epsilon$ of the desired target, we have a recipe for figuring out what error tolerance $T(\epsilon)$ we must achieve for the inputs (i.e. $x_0 \pm T(\epsilon)$) to obtain the desired level of accuracy in the outputs.

ADDITIONAL EDIT:

Of course, such simplistic examples are not representative of what engineers will do in the real world. The kinds of examples one can present in an intro calculus class are necessarily toy examples. You have to crawl before you can walk, and walk before you can run. Here is an example (found by ChatGPT!) of "running":

https://www.mdpi.com/2076-3417/13/8/4999?utm_source=chatgpt.com

As a satellite’s critical load-bearing structure, the large-scale space deployable mechanism (LSDM) is currently assembled using ground precision constraints, which ignores the difference between the ground and space environments. This has resulted in considerable service performance uncertainties in space. To improve satellite service performance, an assembly error model considering the space environment and a tolerance dynamic allocation method based on as-built data are proposed in this paper. Firstly, the factors influencing the service performance during ground assembly were analyzed. Secondly, an assembly error model was constructed, which considers the influence factors of the ground and space environment. Thirdly, on the basis of the assembly error model, the tolerance dynamic allocation method based on as-built data was proposed, which can effectively reduce the assembly difficulty and cost on the premise of ensuring service performance. Finally, the proposed method was validated in an assembly site, and the results show that the pointing accuracy, which is the core indicator of the satellite service performance, was improved from 0.068° to 0.045° and that the assembly cost was reduced by about 13.5%.

A casual glance through the paper will show a much more sophisticated setup and set of approaches, but the basic goal is still to figure out what kind of control of some "input variables" is needed to achieve a desired control of some "output variables". This is epsilon/delta.

One frequently occurring task in engineering is to decompose one complex problem into a bunch of smaller simpler problems.

The epsilon-delta definition allows one to do this, because it allows to chain together many tiny steps to obtain one big complex result in the end.

Suppose that you have three statements similar to that in your question:

• $\forall \color{red}{\epsilon > 0} . \exists \color{green}{\delta > 0} .\forall x_1, x_2.|x_1 - x_2| <\delta \to |f(x_1) - f(x_2)| < \epsilon$
• $\forall \color{green}{\delta > 0} . \exists \color{blue}{\gamma > 0} .\forall y_1, y_2.|y_1 - y_2| <\gamma \to |x(y_1) - x(y_2)| < \delta$
• $\forall \color{blue}{\gamma > 0}.\exists \color{magenta}{\beta >0}. \forall z_1,z_2.|z_1 - z_2| < \beta \to |y(z_1) - y(z_2)| < \gamma$

Then you can nicely compose them all together to infer:

• $\forall\color{red}{\epsilon>0}.\exists\color{blue}{\gamma > 0}.\forall y_1,y_2.|y_1 - y_2|<\gamma\to |f(x(y_1)) - f(x(y_2))| < \epsilon$ (first two)
• $\forall\color{red}{\epsilon>0}.\exists\color{magenta}{\beta > 0}.\forall z_1,z_2.|z_1 - z_2| < \beta \to |f(x(y(z_1))) - f(x(y(z_2)))| < \epsilon$ (all three).

Here, you immediately obtain that the "complex" function $f \circ x \circ y$ is uniformly continuous whenever $f$, $x$ and $y$ are uniformly continuous.

This is because each of those "$\epsilon$-$\delta$" (or "$\gamma$-$\beta$", or whatever) statements acts as a little composable box with inputs ($\epsilon$) and outputs ($\delta$), and you can easily chain multiple such boxes together through their compatible "interfaces".

Moreover, if each of those "existential" statements is actually constructive, i.e. gives you an actual "$\delta$" for each "$\epsilon$" (for simple engineering applications, this is often constructive), then you can literally chain together a bunch of little modular "$\epsilon$-$\delta$" algorithms that will give you the required estimates for the big complex problem.

Without the "$\epsilon$" part, this entire "lego set" becomes useless, because nothing can be composed any more: instead of composable $\epsilon$-$\delta$ "lego bricks" you would give your students a pile of irregularly shaped pebbles, out of which nothing interesting can be built.