Reference

A continuous extension of a function to the entire space is achieved by creating a new function that agrees with the original function on a subset of the domain and is continuous on the entire space. This process involves "filling in the gaps" or smoothing out discontinuities of the original function. The Tietze extension theorem and the Brouwer-Urysohn theorem are important results in this context. The Tietze extension theorem states that any continuous function defined on a closed subset of a normal space can be extended to a continuous function on the entire space. The Brouwer-Urysohn theorem provides an example of a theorem on the existence of a continuous extension of a continuous function, particularly when the subset is closed.

The uniqueness of a continuous extension into a Hausdorff space is also discussed in mathematical literature. If two continuous extensions agree on a dense subset, they must be identical on the entire space. This is due to the fact that the set where they differ would be both open and closed, and in a Hausdorff space, the only such sets are the empty set and the whole space.

In summary, the continuous extension of a function to the entire space is a deep concept in topology and analysis, with various theorems providing conditions under which such an extension is possible and unique. For more detailed explorations, one can refer to resources such as "Uniqueness of a continuous extension of a function into a Hausdorff space" on Math Stack Exchange and "Extensions of functions which preserve the continuity on the original domain" in scientific articles.