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.
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) %}
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 %}.
(see Waldmann pg. 21)