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Differentiability
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differentiable function has a non-Vertical tangent, vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or Cusp (singularity), cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . Differentiability of real functions of one variable A function f:U\subset\mathbb\to\mathbb, defined on an open set U, is ''differentiable'' at a\in U if the derivative :f'(a)=\lim_\frac exists. This implies that the function is continuous ...
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Function Of Several Real Variables
In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ... its applications, a function of several real variables or real multivariate function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... with more than one argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ..., with all arguments being real Real may refer to: * Reality Reality is the s ...
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Tangent Line
In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ..., the tangent line (or simply tangent) to a plane curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ... at a given point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ... is the straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient ...
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Smooth Function
In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ..., the smoothness of a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... is a property measured by the number of continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ... derivatives Derivative may refer to: In mathematics and economics *Brzozowski derivative in the theory of formal languages ...
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Derivative
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the derivative of a function of a real variable In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ... measures the sensitivity to change of the function value (output value) with respect to a change in its argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ... (input value). Derivatives are a fundamental tool of ...
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Continuous Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a continuous function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... such that a continuous variation (that is a change without jump) of the argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ... induces a continuous variation of the value Value or values may refer to: * Value (ethics) In eth ...
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Weierstrass Function
In mathematics, the Weierstrass function is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstrass function has historically served the role of a pathological (mathematics), pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were denounced by contemporaries: Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while Charles Herm ...
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Function (mathematics)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a functionThe words map, mapping, transformation, correspondence, and operator are often used synonymously. . from a set to a set assigns to each element of exactly one element of . The set is called the domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ... of the function and the set is called the codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
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Second Derivative
In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ..., the second derivative, or the second order derivative, of a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... is the derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; f ...
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Intermediate Value Theorem
In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ..., the intermediate value theorem states that if ''f'' is a continuous function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... whose domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ... contains the interval , then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point within ...
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Darboux's Theorem (analysis)
In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the derivative, differentiation of another function has the intermediate value property: the image (mathematics), image of an interval (mathematics), interval is also an interval. When ''ƒ'' is continuously differentiable (''ƒ'' in ''C''1([''a'',''b''])), this is a consequence of the intermediate value theorem. But even when ''ƒ′'' is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be. Darboux's theorem Let I be a closed interval, f\colon I\to \R a real-valued differentiable function. Then f' has the intermediate value property: If a and b are points in I with ay>f'(b). Let \varphi\colon I\to \R such that \varphi(t)=f(t)-yt. If it is the case that f'(a) we adjust our below proof, instead asserting that \varphi has its minimum on [a,b]. Since \va ...
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Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a country located in Central Europe. It is divided into 16 Voivodeships of Pol ... mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ... who is generally considered one of the world's most important and influential 20th-century mathematicians. He was the founder of modern functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat . ...
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Differentiation Rules
This is a summary of differentiation rules, that is, rules for computing the of a in . Elementary rules of differentiation Unless otherwise stated, all functions are functions of that return real values; although more generally, the formulae below apply wherever they are — including the case of .''Complex Variables'', M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, Differentiation is linear For any functions f and g and any real numbers a and b, the derivative of the function h(x) = af(x) + bg(x) with respect to x is : h'(x) = a f'(x) + b g'(x). In this is written as: : \frac = a\frac +b\frac. Special cases include: * ''The ''constant factor rule'' :(af)' = af' * ''The ''sum rule'' :(f + g)' = f' + g' * ''The subtraction rule'' :(f - g)' = f' - g'. The product rule For the functions ''f'' and ''g'', the derivative of the function ''h''(''x'') = ''f''(''x'') ''g''(''x'') with respect to ''x'' is : h' ...
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