Interval package: Difference between revisions

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>

# 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>.