See monotone function on Wiktionary
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Knapp, Basic Real Analysis, Springer, page 357:", "text": "Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere.", "type": "quote" }, { "ref": "2011, Saminathan Ponnusamy, Foundations of Mathematical Analysis, Springer, page 469:", "text": "Monotone functions on #x5B;a,b#x5D; have nice properties. For example, they are integrable on #x5B;a,b#x5D; and have only a countable number of jump discontinuities. In this section, we shall also show that every monotone function is a function of bounded variation, and hence the class #x5C;mathit#x7B;BV#x7D;(#x5B;a,b#x5D;) contains the class of monotone functions on #x5B;a,b#x5D;.", "type": "quote" }, { "ref": "2013, Donald Yau, A First Course in Analysis, World Scientific, page 104:", "text": "We saw in the previous section that monotone functions have some nice properties. For example, a monotone function is continuous except possibly on a countable set. They are also closed under scalar multiplication, and the sum of two increasing functions is increasing. (Exercise (4) on page 103). However, the difference and product of two monotone functions are not necessarily monotone (Exercise (5) on page 103).", "type": "quote" } ], "glosses": [ "A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)." ], "hyponyms": [ { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "nondecreasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotone nondecreasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotonic nondecreasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotonically nondecreasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "increasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotone increasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotonic increasing function" }, { "_dis1": "76 13 11", "sense": "function that never decreases as its independent variable increases", "word": "monotonically increasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "nonincreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotone nonincreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotonic nonincreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotonically nonincreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "decreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotone decreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotonic decreasing function" }, { "_dis1": "77 12 11", "sense": "function that never increases as its independent variable increases", "word": "monotonically decreasing function" } ], "id": "en-monotone_function-en-noun-vrgD6lcF", "links": [ [ "calculus", "calculus" ], [ "function", "function" ], [ "subset", "subset" ], [ "discrete set", "discrete set" ], [ "independent variable", "independent variable" ] ], "raw_glosses": [ "(calculus) A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)." ], "synonyms": [ { "_dis1": "73 13 14", "sense": "function that either never decreases or never increases", "word": "monotonic function" } ], "topics": [ "calculus", "mathematics", "sciences" ], "translations": [ { "_dis1": "49 25 26", "code": "de", "lang": "German", "sense": "calculus: real function that either never decreases or never increases", "tags": [ "feminine" ], "word": "monotone Funktion" }, { "_dis1": "49 25 26", "code": "sh", "lang": "Serbo-Croatian", "sense": "calculus: real function that either never decreases or never increases", "tags": [ "feminine" ], "word": "monotona funkcija" }, { "_dis1": "49 25 26", "code": "tr", "lang": "Turkish", "sense": "calculus: real function that either never decreases or never increases", "word": "monoton fonksiyon" } ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Mathematical analysis", "orig": "en:Mathematical analysis", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "30 32 38", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "31 31 38", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "32 34 33", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "31 31 38", "kind": "other", "name": "Terms with Turkish translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "A function f : X→Y (where X and Y are posets with partial order \"≤\") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x)." ], "id": "en-monotone_function-en-noun-h7rjGnkW", "links": [ [ "mathematical analysis", "mathematical analysis" ], [ "poset", "poset" ], [ "partial order", "partial order" ] ], "raw_glosses": [ "(order theory, mathematical analysis) A function f : X→Y (where X and Y are posets with partial order \"≤\") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x)." ], "topics": [ "mathematical-analysis", "mathematics", "order-theory", "sciences" ] }, { "categories": [ { "_dis": "31 30 40", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "30 31 39", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "30 32 38", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "31 31 38", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "32 34 33", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "29 30 42", "kind": "other", "name": "Terms with Serbo-Croatian translations", "parents": [], "source": "w+disamb" }, { "_dis": "31 31 38", "kind": "other", "name": "Terms with Turkish translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "A Boolean function with the property that switching any one input variable from 0 to 1 results either in no change in output or a change from 0 to 1." ], "id": "en-monotone_function-en-noun-VDXIrdjZ", "links": [ [ "Boolean function", "Boolean function" ] ], "qualifier": "Boolean algebra", "raw_glosses": [ "(Boolean algebra) A Boolean function with the property that switching any one input variable from 0 to 1 results either in no change in output or a change from 0 to 1." ] } ], "synonyms": [ { "_dis1": "36 33 31", "word": "monotonic function" } ], "translations": [ { "_dis1": "40 46 14", "code": "de", "lang": "German", "sense": "order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)", "tags": [ "feminine" ], "word": "monotone Funktion" }, { "_dis1": "40 46 14", "code": "sh", "lang": "Serbo-Croatian", "sense": "order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)", "tags": [ "feminine" ], "word": "monotona funkcija" }, { "_dis1": "40 46 14", "code": "tr", "lang": "Turkish", "sense": "order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)", "word": "monoton fonksiyon" } ], "word": "monotone function" }
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