I have seen it mentioned that in certain infinite dimensional topological vector spaces it is possible to have a smooth curve which is not continuous, but I've never seen an explicit example. Can anybody point me towards a reference for this?
2026-03-31 12:10:17.1774959017
Smooth function which is not continuous
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The mentioner might have been thinking of discontinuous linear maps on infinite-dimensional topological vector spaces. You can read about the construction of such functions at Wikipedia. The construction relies on the axiom of choice.
I am not an expert on notions of differentiability in infinite-dimensional spaces, but any linear function is well-approximated by a linear function (itself!). So I believe that a discontinuous linear function has a Fréchet derivative.
Edit: Christopher Wong explains below that the function will be Gateaux differentiable but not Fréchet differentiable.