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.