Sergei Yakovenko's blog: on Math and Teaching

Properties of continuous functions. Basic notions of topology

The standard list of properties of functions, continuous on intervals, includes theorems on intermediate value, on boundedness, on attainability of extremal values etc.

We explain that these results are manifestations of the following phenomenon. There are several properties of subsets of $\mathbb R^1$ (and, in general, arbitrary subsets of the Euclidean space), which can be defined using only the notions of limit and proximity. Such properties are called topological. Examples of such properties are openness/closeness, connectedness (arc-connectedness) and compactness.

The general principle then can be formulated (vaguely) as follows: the topological properties are preserved by continuous maps (or their inverses).