Can a uniformly continuous function have a weak derivative?. In other words can $C_{unif.~cont.}$ be continuously be embedded in $W^{1,2}(\Omega)$.?
2026-03-26 02:54:34.1774493674
Existence of weak derivative
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Uniformly continuity is not really a very restrictive condition: All continuous functions restricted on $\Omega' \subset \subset \Omega$ is automatically uniform continuous.
To write down a counterexample, consider the Cantor's staircase function $f$, which is a monotone function on $(0,1)$ and is uniform continuous. But $f$ is not in $W^{1, 1}(0,1)$ as it does not satisfy the fundamental theorem of calculus.