Functions of intervals (2) R ∪ {−∞, +∞}. Let f([a, b]) be the image of the bounded interval, equation closed interval [a, b] under the function f. Real analysis result. teaches that if the interval is bounded and the The lattice generated by t function is continuous over the interval, then f([a, b]) y real set, L(IR ∗ ), is the sm is also a closed, bounded interval, and, more significantly, 2 ing IR ∗ that is closed und this extension accomodate [ ] the union of two intervals f([a, b]) = min f(x), max (1) in the lattice can always b x∈[a,b] x∈[a,b] of closed intervals in R ∗ . I Computing the minumum and maximum is trivial if the function is monotonic (Figure 1), and −1/4 also for the non-monotonic standard mathematical functions 0 >>> k = interval([0, x 1] represents the the union [0 The intervals 2 [0, 1], [2, 3], (even-exponent power, cosh, sin, cos...) these are relatively easy to determine. connected components of k sists of only one componen >>> interval[1, 2] interval([1.0, 2.0]) signifying the interval 8[1, 2
Functions of intervals (3) y a f(b) f(a) b x [a, b] −1 =[−∞, +∞] ? [a, b] −1 =[−∞,a −1 ] ∪ [b −1 , +∞] ? 9