“real: chapter_05” — 2011/5/22 — 22:55 — page2—#2
5-2 Real Analysis
When we talk about functions (which are in fact mappings that are
not one-many) whose domain and range are subsets of
R, it is necessary
for us to find out what type of domains are needed for our purposes.
Since most of the concepts such as continuity, differentiability, etc.,
require that the function be defined in a neighbourhood of a point x
as soon as it is defined at x, it is necessary to assume that the domain
of definition should contain a neighbourhood of a point x whenever x
belongs to the domain. In other words, we are forced to study functions
defined on open subsets of a real line. For example, we can define a
function in a domain like
S ={x ∈
R/ 0 < |x − a| <δ}
for some a