Continuous Extension from Dense Subset

Theorem
Let $X$ be a subset of $\R$.

Let $D$ be a dense subset of $X$.

Let $f : D \to \R$ be a continuous mapping.

Let $f$ be uniformly continuous on every  bounded subset of $D$.

Then there exists a unique continuous extension of $f$ to $X$.

Further, this extension is uniformly continuous on every bounded subset of $X$.