This topic concerns a very classical question: *extend of a function between two metric spaces to obtain a new function enjoying certain properties*. I am interested in the following three properties:

- Continuity,
- Uniformly continuity,
- Pointwise equi-continuity, and
- Uniformly equi-continuity.

Throughout this topic, by and we mean metric spaces with metrics and respectively.

**CONTINUITY IS NOT ENOUGH**. Let us consider the first situation where the given function is *only* assumed to be continuous. In this scenario, there is no hope that we can extend such a continuous function to obtain a new continuous function . The following counter-example demonstrates this:

Let and let be any continuous function on such that there is a positive gap between and . For example, we can choose

Since is monotone increasing, we clearly have

Hence any extension of cannot be continuous because will be discontinuous at . Thus, we have just shown that continuity is not enough. For this reason, *we require to be uniformly continuous*.

**SIMPLE OBSERVATIONS**. We start with the following basic results.