Changes

Interval package

, 09:33, 14 October 2014
=== Moore's fundamental theroem of interval arithmetic ===
Let $\mathbf{y} = f(\mathbf{x})$ be the result of
interval-evaluation of $f$ over a box $\mathbf{x} = (x_1,\ldots{},x_n)$
using any interval versions of its component library functions. Then
# In all cases, $\mathbf{y}$ contains the range of $f$ over $\mathbf{x}$, that is, the set of $f(\mathbf{x})$ at points of $\mathbf{x}$ where it is defined: $\mathbf{y} \supseteq \operatorname{Rge}(f \vert \mathbf{x}) = \{ f(\mathbf{x}) \vert x \in \mathbf{x} \cap \operatorname{Dom}(f)\}$
# If also each library operation in $f$ is everywhere defined on its inputs, while evaluating $\mathbf{y}$, then $f$ is everywhere defined on $\mathbf{x}$, that is $\operatorname{Dom}(f) \supseteq \mathbf{x}$.
# If in addition, each library operation in $f$ is everywhere continuous on its inputs, while evaluating $\mathbf{y}$, then $f$ is everywhere continuous on $\mathbf{x}$.
# If some library operation in $f$ is nowhere defined on its inputs, while evaluating $\mathbf{y}$, then $f$ is nowhere defined on $\mathbf{x}$, that is $\operatorname{Dom}(f) = \emptyset$.
== Quick start introduction ==
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