All functions are countable?
115808
Since B is a countable set, one can also associate it with the even and odd natural numbers.  Without loss of generality, map the possibilities of 0->b and 1->b, where b is in B, to the even and odd numbers, respectively.  The union of the even and odds is just the natural numbers, which is countable.
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I have a copy of the second edition. Is that good enough?\n\nIs there something in particular you need, or do you want the outline of the whole book?\n\n1. Set theory\nIntroduction\nFunctions\nUnions, Intersections, and complements\nAlgebra of sets\nThe axiom of chice and infinite direct product\nCountable sets\nRelations and equivalences\nPartial orderings and the maximal principle\nWell ordering and the countable ordinal\n\nPart One -- Theory of Functions of a Real Variable\n\n2. The Real Number System\n\nAxioms for the real numbers\nThe natural and rational numbers as subsets of R\nThe extended real numbers\nSequences of real numbers\nOpen and closed sets of real numbers\nContinuous functions\nBorel sets\n\n3. Lebesgue Measure\nIntroduction\nOuter measure\nMeasurable sets and Lebesgue measure\nA nonmeasurable set\nMeasurable functions\nLittlwood's three principles\n\n4. The Lebesgue Integral\nThe Riemann integral\nThe Lebesgue integral of a bounded function over a set of finite measure\nThe integral of a nonnegative function\nThe general Lebesque integral\nConvergence in measure\n\n5. Differentiation and Integration\nDifferentiation of a monotone function\nFunction of bounded variation\nDifferentiation of an integral\nAbsolute continuity\nConvex functions\n\n6. The Classical Banach Spaces\nThe Lp spaces\nThe Holder Minkowski inequalities\nConvergence and completeneww\nBounded linear functionsals on the Lp spaces\n\n\nPart Two -- Abstract Spaces\n\n7. Metric Spaces\nIntroduction\nOpen and closed sets\nContinuous function and homeomorphisms\nConvergence and completeness\nUniform continuity and uniformity\nSubspaces\nBaire category\n\n8. Topological Spaces\nFundamental notions\nBases and countability\nThe separation axioms and continuous real-valued functions\nProduct spaces\nConnectedness\nAbsolute (G delta)\nNets\n\n9. Compact Spaces\nBasic properties\nCountable compactness and the Bolzano-Weierstrass property\nCompact metric spaces\nProducts of compact spaces\nLocally compact spaces\nThe Stone-Cech compactification\nThe Stone-Weierstrass theorem\nThe Ascoli theorem\n\n10. Banach Spaces\nIntroduction\nLinear Operators\nLinear Functionals and the Hahn-Banach theorem\nThe closed graph theorem\nTopological vector spaces\nWeak topologies\nConvexity\nHilbert space\n\nPart Three -- General Measure and Integration Theory\n\n11. Measure and Integration\nMeasure spaces\nMeasurable functions\nIntegration\nGeneral convergence theorems\nSigned measures\nThe Radon-Nikodym theorem\nThe Lp spaces\n\n12. Measure and Outer Measure\nOuter measure and measurability\nThe extension theorem\nThe Lebesgue-Stieltjes integral\nProduct measures\nInner measure\nExtension by sets of measure zero\nCaratheodory outer measure\n\n13. The Daniell Integral\nIntroduction\nThe extension theorem\nUniqueness\nMeasureability and measure\n\n14. Measure and Topology\nBaire sets and Borel sets\nPositive linear functionals and Baire measure\nBounded linear fuctionals on C(X)\nThe Borel extension of a measure\n\n15. Mappings of Measure Spaces\nPoint mappings and set mappings\nMeasure algebras\nBorel equivalences\nSet mappings and point mappings ofn complete metric spaces\nThe isometries of Lp
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Just substitute x+1 into f(x)\n\nf(x+1) = [ 6(x+1)^3 ] - [ 3(x+1)^2 ] + 4(x+1) - 9
31079
The set of ALL natural numbers is countable by definition, so the equality inherently can only hold for countable values of n, because n was defined to be a natural number, and finite sets are also countale.
679041
acording to my calc teache there are thousands just type it into google and u will find tonz there are 4 main ones
891165
It is COUNTIF and not COUNTA, but with some changes in its parameters\nI didn't find all your options here, but it goes like this:\nto search for all E-01 to E-10 paste this:\n=COUNTIF( A1:A80, "E-??")\nto search both WO1-CW5 along with E-5, paste this\n=COUNTIF( A1:A80, "E-5") + COUNTIF( A1:A80, "WO1-CW5")\n\nYou can also customize it and use it inside SUM function\n=SUM( COUNTIF( blablabla ), COUNTIF( blablabla ))\n\nand there are more usages and functions to do it.\n\nThis is what I got from your question, you didn't type more details, if you feel you want to mail me, do here in Y! Answers\n\nEnjoy my profile, I am the VBAXLMan
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basically, think of a function as a machine:\n\ntake this an an example: f(x) = 2x +3\nsee the x in brackets?\n\nif x = 1, then f(1) = 2(1) + 3 = 5\nif x = 2, then f(2) = 2(2) + 3 = 7\nif x = 3, then f(3) = 2(3) + 3 = 9\n\nsee whats happening? the answer DEPENDS on number inputted in the brackets.\n\nso therefore, the definition you quoted "a consistent, predictable, relationship between the independant and dependant variable" is true.
295853
It would be countable and uncountable *sets*, not numbers.\n\nWe say that a set is countable if it is in one-to-one correspondence with some subset of the set of natural numbers (1,2,3,...). The collection of rational numbers turns out to be countable.\n\nWe say that a set is uncountable if it is not countable. ;) The set of real numbers turns out to be uncountbale, but there are many, many sets that are even larger.\n\nWe say that a set A is a subset of the set B if every element of A is in B, but not necessarily the other way around. So the set of natural numbers form a subset of the set of real numbers.\n\nA tensor is a multilinear map on a vector space. Way too much theory for this space.\n\nGiven a *function* f, an inverse function g, is one where the compositions fog and gof are both the identity map.\n\nThe gap between +ve and -ve infinity makes no real sense unless you simple mean the set of real numbers.
930668
I don't feel like doing the review, but I can offer something.  It turns of that there are functions can are LeBesgue (sp?) integrable but no Riemann.  The f(x) = 0 x rational, =1 x irrational is an example.  You can cook up a set that lands on just the rationals so that it has a limit at a given rational, and hence every rational even though both the rationals and irrationals are infinitely dense in any small neighborhood of any point.  I don't think that you can do the same with irrationals.  If I'm correct then the rationals would be continuous and the irrationals not.  But I don't think that I'm correct.\n\nYou are essentially talking about what's known as point set topology, a part of an undergraduate, or graduate, real analysis course.\n\nI've found Wikipedia to be very helpful for math concepts.\n\nI just found something that said\n\nf(x) = 0, x irrational\nf(x) = x, x rational\n\nis continuous at one point, x = 0.
694927
What are you trying to do?  You have just defined f(x).  You have not said what to do with it.
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It's a recursive function that's not primitive recursive.\n\nIt grows extremely fast.  See the Wikipedia entry.
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Think about your order of operations. If you were evaluting the function for some number, what would you do first? That function is your *inside* function.
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The others are correct:  3x is one such example.\n\nIf you want to be pedantic about it, there are an infinite number of functions that fulfill this.  Just add any constant:\ny=3x+c.
459268
are we counting the whole global???
296652
Where's the function?
607484
Undefined - not infinite
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A function is a set of ordered pairs of numbers, no two of which have the same first element.