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Mathematical function whose derivative exists
continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section Differentiability classes)
Differentiable_function
Degree of differentiability of a function or map
value function f ( x ) = | x | {\displaystyle f(x)=|x|} has class C 0 {\displaystyle C^{0}} , because it is continuous, but not differentiable. Generally
Smoothness
Function defined by multiple sub-functions
that the value of the right sub-function is used in this position. For a piecewise-defined function to be differentiable on a given interval in its domain
Piecewise_function
Function that is continuous everywhere but differentiable nowhere
Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere
Weierstrass_function
Complex-differentiable (mathematical) function
mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each
Holomorphic_function
Theorem in mathematics
versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces
Inverse_function_theorem
On converting relations to functions of several real variables
Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions". Calculus of Several Variables and Differentiable Manifolds. New York: Macmillan. pp. 54–88
Implicit_function_theorem
Differentiable function whose derivative is not Riemann integrable
properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. The function is defined by
Volterra's_function
Manifold upon which it is possible to perform calculus
another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold
Differentiable_manifold
Real function with secant line between points above the graph itself
that interval. If a function is differentiable and convex then it is also continuously differentiable. A differentiable function of one variable is convex
Convex_function
Theorem in real analysis
analysis, Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary
Rolle's_theorem
Instantaneous rate of change (mathematics)
derivatives are the result of differentiating a function repeatedly. Given that f {\displaystyle f} is a differentiable function, the derivative of f {\displaystyle
Derivative
Mathematical operation in calculus
can be found by similar methods. Let F {\displaystyle F} be a differentiable function of two variables, and suppose that an equation F ( x , y ) = f
Implicit_differentiation
Mathematical functions which are smooth but not analytic
In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its domain
Non-analytic_smooth_function
Property of a mathematical function
zero (note that this indicator function is not left differentiable at zero). If a real-valued, differentiable function f, defined on an interval I of
Semi-differentiability
Mathematical relation consisting of a multi-variable function equal to zero
an implicit function that is differentiable in some small enough neighbourhood of (a, b); in other words, there is a differentiable function f that is defined
Implicit_function
Strong form of uniform continuity
to 1. Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable The function f ( x ) = { x 2 sin ( 1 /
Lipschitz_continuity
Study of rates of change
approximation to a differentiable function near a point. In this sense, differentiation is closely related to the differential. For functions of several variables
Differential_calculus
Measure of local oscillation behavior
C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} is the set of continuously differentiable vector functions of compact support contained in Ω {\displaystyle \Omega }
Total_variation
Derivative defined on normed spaces
function that is Fréchet differentiable at a point is necessarily continuous there and sums and scalar multiples of Fréchet differentiable functions are
Fréchet_derivative
Rules for computing derivatives of functions
{d^{k}}{dx^{k}}}g(x).} Differentiable function – Mathematical function whose derivative exists Differential of a function – Notion in calculus Differentiation of integrals –
Differentiation_rules
Type of function in mathematics
complex differentiable at every point of the set. For this reason, in complex analysis the terms analytic function and holomorphic function are often
Analytic_function
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Analyzes the topology of a manifold by studying differentiable functions on that manifold
by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold
Morse_theory
Mathematical process of finding the derivative of a trigonometric function
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
Nowhere analytic, infinitely differentiable function
the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). This function satisfies
Fabius_function
Point where the derivative of a function is zero or undefined (in certain cases)
Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian
Critical_point_(mathematics)
Multivariate derivative (mathematics)
of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle
Gradient
Branch of mathematics studying functions of a complex variable
holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic function is always infinitely
Complex_analysis
Approximation of a function by a polynomial
Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k}
Taylor's_theorem
Theorem in mathematics
theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to
Mean_value_theorem
Matrix of partial derivatives of a vector-valued function
be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist. If f is differentiable at
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Operation in differential calculus
differentiable at x = 0, but is symmetrically differentiable there with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does
Symmetric_derivative
Notion in calculus
and differentiable functions f and g, d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.} Product rule: For two differentiable functions
Differential_of_a_function
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Association of one output to each input
century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized
Function_(mathematics)
Differentiable function whose derivative is everywhere injective
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly
Immersion_(mathematics)
Counterintuitive mathematical object
Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is
Pathological_(mathematics)
Negative of a convex function
a\}} are convex sets. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically
Concave_function
Indefinite integral
function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function
Antiderivative
About maxima and minima of functions
non-differentiable: f is not differentiable at x 0 {\displaystyle x_{0}} stationary point: f ′ ( x 0 ) = 0 {\displaystyle f'(x_{0})=0} The function f
Interior_extremum_theorem
Generalized function whose value is zero everywhere except at zero
delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure
Dirac_delta_function
Formula in calculus
formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives of z and y. More precisely
Chain_rule
Type of derivative in mathematics
{\displaystyle a} , then f {\displaystyle f} is differentiable at a {\displaystyle a} . If f {\displaystyle f} is differentiable at a point, then the derivative of
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Mathematical function with no sudden changes
everywhere continuous but nowhere differentiable. The derivative f ′ ( x ) {\displaystyle f'(x)} of a differentiable function f ( x ) {\displaystyle f(x)}
Continuous_function
Evaluates a line integral through a gradient field using the original scalar field
rather than just the real line. If φ : U ⊆ Rn → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point
Gradient_theorem
Method of mathematical differentiation
implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero. The method is used
Logarithmic_differentiation
Generalization of the concept of directional derivative
redirect targets Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Differentiation in Fréchet spaces –
Gateaux_derivative
exist functions which are not in any Baire class. Examples: The derivative of any differentiable function is of class 1. An example of a differentiable function
Baire_function
Method in statistics
when the random variable being considered can be defined as a differentiable function of a random variable which is asymptotically Gaussian. More generally
Delta_method
Concept in mathematical analysis
derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense
Pompeiu_derivative
Mathematics of real numbers and real functions
degrees of regularity. A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having derivatives
Real_analysis
Continuously differentiable function: differentiable, with continuous derivative. Smooth function: Has derivatives of all orders. Lipschitz function, Holder
List_of_types_of_functions
Mathematical idealization of the trace left by a moving point
regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane
Curve
Description of continuous random distribution
density function if its cumulative distribution function F(x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its
Probability_density_function
Theorem in convex analysis
derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri
Danskin's_theorem
Order-preserving mathematical function
{\displaystyle f} is a monotonic function defined on an interval I {\displaystyle I} , then f {\displaystyle f} is differentiable almost everywhere on I {\displaystyle
Monotonic_function
Polynomial function of degree 3
values of a function and its derivative at some sampling points, one can interpolate the function with a continuously differentiable function, which is
Cubic_function
Calculus of vector-valued functions
are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function f(x, y) with real values, one
Vector_calculus
Zero of the derivative of a function
a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally
Stationary_point
Second-order partial differential equation describing motion of mechanical system
the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains
Euler–Lagrange_equation
Theorem in analysis
versions of the Wirtinger inequality: Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) =
Wirtinger's inequality for functions
Wirtinger's_inequality_for_functions
Differential equation that is linear with respect to the unknown function
arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable
Linear_differential_equation
Mathematical model of the time dependence of a point in space
equations. If Φ is continuously differentiable the system is called a differentiable dynamical system. The function f is therefore a "smooth" mapping
Dynamical_system
Objects that generalize functions
is not shared by most other notions of differentiation. If m : U → R is an infinitely differentiable function and T is a distribution on U, then the product
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Technique in integral evaluation
requirement that φ be continuously differentiable can be replaced by the weaker assumption that φ be merely differentiable and have a continuous inverse.
Integration_by_substitution
Method for finding stationary points of a function
Newton–Raphson) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x
Newton's method in optimization
Newton's_method_in_optimization
Isomorphism of differentiable manifolds
continuously differentiable. Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a continuously differentiable map f : M →
Diffeomorphism
Method of differentiating single-term polynomials
differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the
Power_rule
Mathematical transformation
of differentiable manifolds). This definition is equivalent to the modern mathematicians' definition as long as f {\displaystyle f} is differentiable and
Legendre_transformation
Characteristic property of holomorphic functions
Conversely, if the functions u and v are (real) differentiable at z and satisfy the Cauchy-Riemann equations there, then f is complex-differentiable at z. In this
Cauchy–Riemann_equations
In vector calculus, an invex function is a differentiable function f {\displaystyle f} from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb
Invex_function
Largest and smallest value taken by a function at a given point
points on the boundary, and take the greatest (or least) one. For differentiable functions, Fermat's theorem states that local extrema in the interior of
Maximum_and_minimum
Programming paradigm
Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation
Differentiable_programming
Numerical calculations carrying along derivatives
to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation enables the simultaneous computation of the
Automatic_differentiation
Type of infinitesimal in calculus
for some differentiable function Q {\displaystyle Q} in an orthogonal coordinate system (hence Q {\displaystyle Q} is a multivariable function whose variables
Exact_differential
Function with a multiplicative scaling behaviour
every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k, defined on a
Homogeneous_function
Mapping which preserves all topological properties of a given space
\theta \right).} The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is a
Homeomorphism
Mathematical method in calculus
v} to be continuously differentiable. Integration by parts works if u {\displaystyle u} is absolutely continuous and the function designated v ′ {\displaystyle
Integration_by_parts
Concept in mathematics
iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative
Mirror_descent
Mathematical function, denoted exp(x) or e^x
definitions of the exponential function, although of very different nature. The exponential function is the unique differentiable function that equals its derivative
Exponential_function
Topological space that locally resembles Euclidean space
additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric
Manifold
All derivatives have the intermediate value property
→ R {\displaystyle f\colon I\to \mathbb {R} } be a real-valued differentiable function. Then f ′ {\displaystyle f'} has the intermediate value property:
Darboux's_theorem_(analysis)
Theorem of convex functions
building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears
Jensen's_inequality
Form of projection
\mathbb {R} ,\ i=1,\dots ,n} are possibly non-differentiable convex functions. The lack of differentiability rules out conventional smooth optimization techniques
Proximal_gradient_method
Optimization algorithm
It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in the opposite direction
Gradient_descent
Mathematical theorem
Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure
Rademacher's_theorem
Real function with finite total variation
chains of inclusions for continuous functions over a closed, bounded interval of the real line: Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely
Bounded_variation
Concept in complex analysis
variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit
Wirtinger_derivatives
Mathematical theorem, used in calculus
f^{-1}(z)+C.} Because all holomorphic functions are differentiable, the proof is immediate by complex differentiation. Mathematics portal Integration by
Integral_of_inverse_functions
Function whose composition with the logarithm is convex
X, then it vanishes everywhere in the interior of X. If f is a differentiable function defined on an interval I ⊆ R, then f is logarithmically convex
Logarithmically convex function
Logarithmically_convex_function
Oscillatory error in Fourier series
continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing
Gibbs_phenomenon
requiring x ≠ y {\displaystyle x\neq y} . A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the
Strict_differentiability
Differentiable function in functional analysis
discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector
Differentiable vector-valued functions from Euclidean space
Differentiable_vector-valued_functions_from_Euclidean_space
inverse function see also List of differential geometry topics Differentiable manifold Differentiable structure Submersion – a differentiable map between
List_of_real_analysis_topics
Disproved conjecture
operation of transposition, f(e) is scalar function, and f(0) = 0. Suppose, f(e) is a differentiable function and the following condition k 1 < f ′ ( e
Kalman's_conjecture
Type of function
its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has
Pseudoconvex_function
Algorithm for finding zeros of functions
Suppose that the function f has a zero at α, i.e., f(α) = 0, and f is differentiable in a neighborhood of α. If f is continuously differentiable and its derivative
Newton's_method
Point where the curvature of a curve changes sign
the curvature changes its sign. For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative
Inflection_point
Theorem of probability theory
variable with expectation μ and variance σ2. Further suppose g is a differentiable function for which the two expectations E ( g ( X ) ( X − μ ) ) {\displaystyle
Stein's_lemma
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