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Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $$F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$$ is this extension which is uniformly continuous on the whole $X$.

Yes, and even more is true. The argument is as follows: let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|_K$ is uniformly continuous; let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits a uniformly continuous extension to $X$ with the same modulus - more concretely, $$F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$$ is this extension which is uniformly continuous on the whole $X$.

Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$ is this extension which is uniformly continuous on the whole $X$.

Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $$F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$$ is this extension which is uniformly continuous on the whole $X$.

Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$ is this extension which is uniformly continuous on the whole $X$.

Yes, even more is true. The argument is as follows. Let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact. Then $f|_K$ is uniformly continuous, let $\omega$ be its nondecreasing subadditive modulus of continuity. By McShane-Whitney's extension formula $f|_K$ admits an uniformly continuous extension to $X$ with the same modulus, more concretely $F(x)=\inf\{f(k)+\omega(d(k,x))\colon k\in K\}$ is this extension which is uniformly continuous on the whole $X$.