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Galileo’s discovery is that from these two simpler motions, in the plane and on the line, he could completely recapture the complicated motion in space. In fact, if the motions of the shadow and the level are ‘continuous’, so that the shadow does not suddenly disappear from one place and instantaneously reappear in another...

  • Conceptual Mathematics, A first introduction to Categories by Lawvrere

I'm trying to understand the above paragraph. So, far, it's clear to me that the continuity considered here can be taken as continuity of a curve in $\mathbb{R^4}$ or $\mathbb{R^3}$ (seen as a map from interval of time), and that of the projection of the curve in two subspaces, but why does continuity in two subspaces here means continuity in the whole space?

I checked on MSE and it seems that it is generally not a true fact see eg. So, what additional assumptions do we need for continuity in slots to make continuity as a pair?

tryst with freedom
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    The MSE link talks about the situation when the domain is a product space. You’re considering curves, so the domain is simply an interval $I\subset\Bbb{R}$, which is not a product space. It is your target space which could be considered as a product ($\Bbb{R}^4\cong\Bbb{R}\times\Bbb{R}^3\cong\Bbb{R}\times\Bbb{R}\times\Bbb{R}\times\Bbb{R}$, after fixing a frame of reference). In this case, continuity of a map into a product target space is equivalent to each mapping being continuous (almost by definition of the product topology). Anyway, what does any of this have to do with physics? – peek-a-boo Jul 04 '23 at 11:22

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