Examples of functions $^$-extendable for each  finite, but not $^∞$-extendable

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Instytut Matematyki, Uniwersytet Jagielloński,, ul. Reymonta 4, 30-059 Kraków, Poland,

In Example 1, we describe a subset X of the plane and a function on X which has a

^k

-extension to the whole

ℝ^{2}

for each finite, but has no

^\infty

-extension to

ℝ^{2}

. In Example 2, we construct a similar example of a subanalytic subset of

ℝ^{5}

; much more sophisticated than the first one. The dimensions given here are smallest possible.

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