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