Dilek

You have identified something real. The honest description of what is happening has three steps, and only the middle one is calculus.

1. Somebody chose a continuous extension. The formula for delay or cost was written using operations that happen to make sense for any real number, so it defines a continuous function that agrees with the discrete one at the integers.
2. The extension is optimised with ordinary calculus.
3. The result is rounded back to an integer.

Step 1 is the part that is never stated and it is where the assumption lives. There is no unique continuous extension of a function on the integers — infinitely many curves pass through the same set of points, and they have their minima in different places. The one being used is whichever one the algebra happened to produce.

So it is not illegitimate, but it is a modelling choice being presented as a derivation.

The structural answer is that the problem has no useful local structure to exploit.

Compare it with something easy. Finding the shortest path between two points on a network is easy because of a property that makes the whole thing collapse: any sub-path of a shortest path is itself a shortest path. That lets you build the answer from smaller answers and never revisit.

The salesman problem has no equivalent property. A tour that visits a subset of cities optimally may not appear in the optimal full tour at all, because the full tour has to enter and leave that subset at particular points, and those entry and exit choices change what the best sub-tour is. Optimal pieces do not assemble into an optimal whole.

Once that fails, you have lost the technique that makes essentially every easy optimisation problem easy, and what remains is search over a space of routes that grows faster than exponentially with the number of cities.

The popular science attention has a third, more mundane source: it produces results that are reliably counter-intuitive, which makes for good writing.

The standard example is a screening test that is 99% accurate for a condition affecting one person in ten thousand. Most people's instinct is that a positive result means you almost certainly have it. The theorem says it is under one percent likely, because the enormous number of healthy people generates far more false positives than there are true cases.

That gap between intuition and answer is large, the arithmetic fits on a napkin, and the conclusion matters — it applies to medical screening, security alerting, fraud detection and spam filtering. A result that is simple, surprising and consequential is going to get books written about it regardless of how hard the proof was.

The other half is a counting argument that I find persuasive even though it is informal.

"Closed form" means expressible as a finite combination of a small fixed set of functions — powers, exponentials, logarithms, trigonometric functions and a few others. That set is countable: you can in principle list every such expression.

The space of solutions to differential equations is vastly larger than that. So most solutions cannot possibly have a closed form, for the same reason most real numbers are not describable by any finite formula. There simply are not enough formulas to go around.

From this angle the surprise is inverted. The question is not why so few equations are solvable, but why any are — and the answer is that the ones in textbooks were selected precisely because they are, which is what creates the misleading impression in the first place.