We are analyzing several FTC proofs to gain some insights. For now it seems all of them need analysis to be stated with enough detail to be convincing.

In this proof, I stumbled upon an assumption that can be reduced to claiming that this statement is true:

As usual, this is intuitively very true, the interval vanishes, leaving the 'sup' to act on only 'one point' if f is continuous. But that is no proof.

I have tried to detail this a bit more to see if I can prove it, the main idea behind my proof is:

I have found courage to be an essential component across many proofs and bold inventions in mathematics.

I am not sure how good it is, it feels pretty convincing, but there are 2 spots where it needs more detail, and I suspect that for these spots, there is an inescapable need for analysis (luckily we will be tackling that in the foreseeable future).

**Courage**.I have found courage to be an essential component across many proofs and bold inventions in mathematics.

I am not sure how good it is, it feels pretty convincing, but there are 2 spots where it needs more detail, and I suspect that for these spots, there is an inescapable need for analysis (luckily we will be tackling that in the foreseeable future).

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