Continuous Mapping (Metric Space)/Examples

Examples of Continuous Mappings in the Context of Metric Spaces

Let the following mappings be defined:

$\ds g: \R^2 \to \R^2 \times \R^2: \, $

$\ds \map g {x, y}$

$=$

$\ds \tuple {\tuple {x, y}, \tuple {x, y} }$

$\ds h: \R^2 \times \R^2 \to \R \times \R: \, $

$\ds \map h {\tuple {a, b}, \tuple {c, d} }$

$=$

$\ds \tuple {a + b, c - d}$

$\ds k: \R \times \R \to \R \times \R: \, $

$\ds \map k {u, v}$

$=$

$\ds \tuple {u^2, v^2}$

$\ds m: \R \times \R \to \R: \, $

$\ds \map k {x, y}$

$=$

$\ds \dfrac {x - y} 4$

where $\R$ and $\R^2$ denote the real number line and real number plane respectively, under the usual (Euclidean) metric.

Then:

each of $g, h, k, m$ are continuous

$x y = \map {\paren {m \circ k \circ h \circ g} } {x, y}$

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Then $f$ is not continuous.