[tgv]

Inequalities didn't make sense to me. It's harder to understand than the delta-epsilon from the opening, and it doesn't feel like it translates back to a better understanding.

Convergence is presented as just a pattern. It doesn't have to be economic, but the example naturally suggests convergence, so that's ok.

But continuity and differentiability didn't make sense either. You don't "buy" continuity. There's no (increasing) value attached to smaller intervals, at least not in my understanding of it.

[thaumasiotes]

The metaphor in the post says that, if you have a continuous function and you want to restrict its value to within a very small range, you "pay" for that by restricting the value of the independent variable to a suitably small range. That such a "payment" is possible [phrased another way, that this payment will have the effect you want] is what it means for the function to be continuous.

[tgv]

Numerically it makes sense, but I don't have the feeling of cost at all with range restriction. If anything, it should become cheaper.

So, I'm way, way below the Olympic status of Terry Tao, but he might be abstracting a bit too much here. This may not help students understand the topic.

[jerf]

People need different metaphors. So it's often good to present students with a slate of them; if one doesn't work, try another.

It's important not to get stuck on the metaphor, but in practice all but the weirdest of us need them to bootstrap into a mathematical intuition.

This one does absolutely nothing for me either; even casting my mindset back to when I first encountered the episilon-delta treatement I don't think this would have helped me. But if it helps others, that's great.

Also, I think this is a concise treatment. If I were to try to present this to a math class, I'd expand it into at least half a class session, if not a full one. Terry Tao is presenting the metaphor fairly directly, not for pedagogical purposes on this post itself. If 3Blue1Brown took this post and ran with it I'm sure a lot more people would find the result useful at that density of presentation.

[int3]

Increased precision typically costs more economically, so I think it's a pretty good analogy... precise physical measurements require specialized equipment, precise floating-point calculations require more computational power, etc