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According to Cardinality of set of real continuous functions the set of all continuous functions is $|\mathbb{R}|$ using similar reasoning the set of continuous functions within a ball is also bijective to the reals.

Because of this you can create a function $U :: \mathbb{R} \to (\mathbb{R} \to \mathbb{R})$. It is not difficult to "repair" any discontinuities in such a function so it will be continuous. By the reasoning given in this paper https://arxiv.org/pdf/math/0305282.pdf all functions in this system must be degenerate; that is they must all have fixed points thus proving Brouwer fixed-point theorem.

Q the Platypus
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    As far as I know, no approach to the Brouwer fixed point theorem via the Lawvere fixed point theorem is known; I asked about this on MO years ago (https://mathoverflow.net/questions/136478/can-the-lawvere-fixed-point-theorem-be-used-to-prove-the-brouwer-fixed-point-the). Nobody knows how to construct a suitable function $U$ and you haven't written down any kind of argument that it can be done. The cardinality argument doesn't tell you anything about whether you can make $U$ continuous. – Qiaochu Yuan Sep 16 '23 at 06:15
  • My thought was to create a continuous function by taking a discontinuous function and then fixing any discontinuities by creating a Homotopy between both sides and effectively bridging the gap after reading the references I realise this isn’t as trivial as I thought. – Q the Platypus Sep 16 '23 at 11:43

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