Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any known result guaranteeing that $f$ can be extended to a uniformly continuous function $F:D(0,M)\rightarrow Y$?
Extension Preserving Modulus of Continuity
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