Continuous Function - Continuous Functions Between Metric Spaces

Continuous Functions Between Metric Spaces

The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set X equipped with a function (called metric) d_X, that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function

that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces (X, d_X) and (Y, d_Y) and a function

then f is continuous at the point c in X (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying d_X(x, c) < δ will also satisfy d_Y(f(x), f(c)) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (x_n) in X with limit lim x_n = c, we have lim f(x_n) = f(c). The latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (x_n) in X with limit c, the sequence (f(x_n)) is a Cauchy sequence, and c is in the domain of f.

The set of points at which a function between metric spaces is continuous is a G_δ set – this follows from the ε-δ definition of continuity.

This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator

between normed vector spaces V and W (which are vector spaces equipped with a compatible norm, denoted ||x||) is continuous if and only if it is bounded, that is, there is a constant K such that

for all x in V.