On the uniform approximation of Cauchy continuous functions

In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions - the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point - are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space.

Two classes of metric spaces

The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

Rapid identification and differentiation of Mycobacterium avium subspecies paratuberculosis types by use of real-time PCR and high-resolution melt analysis of the MAP1506 locus

High-resolution melt (HRM) analysis can identify sequence polymorphisms by comparing the melting curves of amplicons generated by real-time PCR amplification. We describe the application of this technique to identify Mycobacterium avium subspecies paratuberculosis types I, II, and III. The HRM approach was based on type-specific nucleotide sequences in MAP1506, a member of the PPE (proline-proline-glutamic acid) gene family.