Continuity
Overview
The notion of a continuous map is foundational to mathematical
analysis.
Its definition has evolved over time. For most analysis, the
limit definition (see below) is sufficient, but the current topological
definition encompasses the limit definition.
Topological Definition
A map
{% f:M \rightarrow N %}
where M and N are both topological spaces. Then, f is continuous if the pre-image {% f^{-1}(B) %} is open in M whenever
B is open in N.
Limit Definition
A function f, is said to be continuous if the
limit
of the function as its argument approaches some point in the domain is the value of the function at that point.
{% \lim_{x \rightarrow x_0} f(x) = f(x_0) %}
Topics
