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Function which is not continuous at any point of its domain
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its
Nowhere_continuous_function
Function that is continuous everywhere but differentiable nowhere
differentiable nowhere. It is also an example of a fractal curve. The Weierstrass function has historically served the role of a pathological function, being
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Mathematical function whose derivative exists
continuously differentiable if its derivative is also a continuous function over the domain of f {\textstyle f} . Continuous functions may be nowhere
Differentiable_function
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
equal to f (x) + f (y). Continuous function: in which preimages of open sets are open. Nowhere continuous function: is not continuous at any point of its
List_of_types_of_functions
Degree of differentiability of a function or map
function has all derivatives up to order k {\displaystyle k} , and such that all of these derivatives are continuous. One says that such a function has
Smoothness
provides a sort of maximally discontinuous linear map (confer nowhere continuous function). Note that X is not complete here, as must be the case when
Discontinuous_linear_map
rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous
List of mathematical functions
List_of_mathematical_functions
Uniform restraint of the change in functions
In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle
Uniform_continuity
In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits
Baire_function
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Indicator function of rational numbers
example of a pathological function which provides counterexamples to many situations. The Dirichlet function is nowhere continuous. We can prove this by reference
Dirichlet_function
Branch of mathematics
of real functions. Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling
Mathematical_analysis
Form of continuity for functions
interval and is nowhere zero then 1/f is absolutely continuous. Every absolutely continuous function (over an interval) is uniformly continuous and, therefore
Absolute_continuity
Function that is discontinuous at rationals and continuous at irrationals
{\displaystyle f} is continuous at every irrational number, so its points of continuity are dense within the real numbers. f {\displaystyle f} is nowhere differentiable
Thomae's_function
Any real function on R admits a continuous restriction on a dense subset of R
{\displaystyle \mathbb {Q} } is continuous, although the Dirichlet function is nowhere continuous in R . {\displaystyle \mathbb {R} .} More generally, a Blumberg
Blumberg_theorem
a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the
Quasi-continuous_function
Generalized function whose value is zero everywhere except at zero
called the delta function because it is a continuous analogue of the Kronecker delta function. The mathematical rigor of the delta function was disputed until
Dirac_delta_function
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)
Mathematics of real numbers and real functions
also degrees of regularity. A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having
Real_analysis
sequence Function of a real variable Real multivariable function Continuous function Nowhere continuous function Weierstrass function Smooth function Analytic
List_of_real_analysis_topics
Subset whose closure is the whole space
function. In other words, the polynomial functions are dense in the space C [ a , b ] {\displaystyle C[a,b]} of continuous complex-valued functions on
Dense_set
"Every separable Banach space is isometric to a space of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. 123 (12). American Mathematical
Banach–Mazur_theorem
Theorem in topology
in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself
Brouwer_fixed-point_theorem
Branch of mathematics studying functions of a complex variable
\mathbb {R} .} A complex function is continuous if and only if its associated vector-valued function of two variables is also continuous. However, this identification
Complex_analysis
Mathematical functions which are smooth but not analytic
}}x\leq 0,\end{cases}}} defined for every real number x. The function f has continuous derivatives of all orders at every point x of the real line. The
Non-analytic_smooth_function
Integral expressing the amount of overlap of one function as it is shifted over another
one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete
Convolution
All derivatives have the intermediate value property
Darboux function even though it is not continuous at one point. An example of a Darboux function that is nowhere continuous is Conway's base 13 function. Another
Darboux's_theorem_(analysis)
cliquish function is similar to, but weaker than, the notion of a continuous function and quasi-continuous function. All (quasi-)continuous functions are cliquish
Cliquish_function
Mathematical theorem in the study of analysis
that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because
Stone–Weierstrass_theorem
Instantaneous rate of change (mathematics)
first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan
Derivative
Mathematical set whose closure has empty interior
2011, Example 11.5.3(f). "Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points with
Nowhere_dense_set
Functions such that f(–x) equals f(x) or –f(x)
function's being odd or even does not imply differentiability, or continuity. For example, the Dirichlet function is even, but is nowhere continuous.
Even_and_odd_functions
Japanese mathematician
field theory. The Blancmange curve, the graph of a nowhere-differentiable but uniformly continuous function, is also called the Takagi curve after his work
Teiji_Takagi
Stochastic process generalizing Brownian motion
negative values on (0, ε). The function w is continuous everywhere, but nowhere differentiable (like the Weierstrass function). For any ϵ > 0 {\textstyle
Wiener_process
Type of function in mathematics
products, and compositions of analytic functions are analytic. The reciprocal of an analytic function that is nowhere zero is analytic. In one variable, the
Analytic_function
Mathematical phrase
This order is a pointed dcpo, where the least element is the nowhere-defined partial function (with empty domain). In fact, ≤ is also bounded complete. This
Complete_partial_order
Topology in the study of subharmonic functions
the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Concepts in the fine topology are normally
Fine topology (potential theory)
Fine_topology_(potential_theory)
Theorem in probability and statistics
possible values x of X. If instead the distribution of X is continuous with probability density function fX, then the expected value of g(X) is E [ g ( X )
Law of the unconscious statistician
Law_of_the_unconscious_statistician
Logarithm of a complex number
real-valued functions, by integration of 1 / z {\displaystyle 1/z} , or by the process of analytic continuation. There is no continuous complex logarithm
Complex_logarithm
Smooth nowhere-analytic function
increasing on [ 0 , 1 ] {\displaystyle [0,1]} but nowhere real-analytic. It is the distribution function of the random series X = ∑ n = 1 ∞ 2 − n U n , {\displaystyle
Fabius_function
Branch of topology
point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are
General_topology
Method of mathematical integration
Consider the indicator function of the rational numbers, 1Q, also known as the Dirichlet function. This function is nowhere continuous. 1 Q {\displaystyle
Lebesgue_integral
Property holding for typical examples
subset in the Cr topology. Here Cr is the function space whose members are continuous functions with r continuous derivatives from a manifold M to a manifold
Generic_property
Fractal curve
shows a variation of the curve as an example of a continuous everywhere yet nowhere differentiable function that was possible to represent geometrically at
Koch_snowflake
Topics referred to by the same term
Lambert W function, a set of functions where w is any complex number Weierstrass function (nowhere-differentiable function), a real function continuous everywhere
W_(disambiguation)
Countable intersection of open sets
] ) {\displaystyle C([0,1])} . (See Weierstrass function § Density of nowhere-differentiable functions.) The notion of Gδ sets in metric (and topological)
Gδ_set
"Small" subset of a topological space
of A {\displaystyle A} , which consists of the continuous real-valued nowhere differentiable functions on [ 0 , 1 ] , {\displaystyle [0,1],} is comeagre
Meagre_set
German mathematician
theory of algebraic functions and on the Riemann–Roch theorem. The Takagi–Landsberg curve, a fractal that is the graph of a nowhere-differentiable but
Georg_Landsberg
Everywhere except a set of measure zero
converges to f almost everywhere. A bounded function f : [a, b] → R is Riemann integrable if and only if it is continuous almost everywhere. As a curiosity, the
Almost_everywhere
Differential form
many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds
Volume_form
Collection of random variables
sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk
Stochastic_process
About mathematical functions
series. Fourier had a general conception of a function, which included functions that were neither continuous nor defined by an analytical expression. Related
History of the function concept
History_of_the_function_concept
Mathematical method in calculus
{\displaystyle v} to be continuously differentiable. Integration by parts works if u {\displaystyle u} is absolutely continuous and the function designated v ′
Integration_by_parts
Vector space with a notion of nearness
operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a vector topology and every topological
Topological_vector_space
Multifractal function used in terrain modeling and simulation
Karl Weierstrass as an example of a continuous but nowhere differentiable function. The Weierstrass–Mandelbrot function builds on this foundation by incorporating
Weierstrass–Mandelbrot function
Weierstrass–Mandelbrot_function
Right inverse of a fiber bundle map
section) of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection function π {\displaystyle \pi } . In other words, if E {\displaystyle
Section_(fiber_bundle)
Czech mathematician (1897–1970)
definition of a continuous function that was nowhere differentiable. Bolzano's 1830 discovery predated the 1872 publication of the Weierstrass function, previously
Vojtěch_Jarník
Distance from zero to a number
interval [−1, 1]. The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations
Absolute_value
Statistical model
many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes
Gaussian_process
Paradox involving infinity
already killed them. The paradox raises questions about the possibility of continuous time and the infinite past (temporal finitism). It is inspired by J. A
Grim_Reaper_paradox
Operator on a Hilbert space that shifts basis vectors
characteristically fractal in shape, often differentiable-nowhere or even continuous-nowhere. Eigenvalues on the unit circle are associated with unitary
Unilateral_shift_operator
Stochastic process modeling random walk with friction
its mean function: such a process is called mean-reverting. The process can be considered to be a modification of the random walk in continuous time, or
Ornstein–Uhlenbeck_process
Determinant of the matrix of first derivatives of a set of functions
implies linear dependence. Peano (1889) pointed out that the functions x2 and |x| · x have continuous derivatives and their Wrońskian vanishes everywhere, yet
Wronskian
Partial differential equation technique
relation, as this is a function in one variable. A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly
Homotopy_principle
Type of mathematical plane curve
support functions based on the Weierstrass function whose corresponding projective hedgehogs are fractal curves that are continuous but nowhere differentiable
Hedgehog_(geometry)
German mathematician (1826–1866)
representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable
Bernhard_Riemann
Differential equations involving stochastic processes
continuous time limit of a stochastic difference equation. In physics, the main method of solution is to find the probability distribution function as
Stochastic differential equation
Stochastic_differential_equation
Fifteen open problems in mathematical physics
Sergey A. (June 2003). "On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm–Liouville
Simon_problems
Cylindrical conformal map projection
the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude
Mercator_projection
Class of computational problems
different commodities whose total flow amounts together respect the capacities Nowhere-zero flow, a type of flow studied in combinatorics in which the flow amounts
Network_flow_problem
Multivariate derivative (mathematics)
integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function. The gradient of a function f : R n → R {\displaystyle
Gradient
topology for which all the projection maps are continuous. Proper function/mapping A continuous function f from a space X to a space Y is proper if f −
Glossary_of_general_topology
System for reasoning about vagueness
sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular
Fuzzy_logic
Metric geometry
{\displaystyle C(a,b)} of continuous functions on ( a , b ) , {\displaystyle (a,b),} for it may contain unbounded functions. Instead, with the topology
Complete_metric_space
Concept in mathematical analysis
boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence
Dirichlet_kernel
Symbol representing a mathematical object
formalized enough to deal with apparent paradoxes such as a nowhere differentiable continuous function. To solve this problem, Karl Weierstrass introduced a
Variable_(mathematics)
Type of mathematical functions
connected set D in C {\displaystyle \mathbb {C} } we can find a function that will nowhere continue analytically over the boundary, that cannot be said for
Function of several complex variables
Function_of_several_complex_variables
Difference between logarithm and harmonic series
Johann von Soldner in 1809, who used the notation H. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later
Euler's_constant
Measure theory
Almost every continuous function from the interval [ 0 , 1 ] {\displaystyle [0,1]} into the real line R {\displaystyle \mathbb {R} } is nowhere differentiable;
Prevalent_and_shy_sets
Interpretation of quantum mechanics
interpretation of quantum mechanics that postulates that, in addition to the wave function, a particle possesses a definite position at all times, even when unobserved
De_Broglie–Bohm_theory
Theorem stating that pointwise boundedness implies uniform boundedness
pointwise convergent sequence of equicontinuous functions on a compact set converges to a continuous function. By uniform boundedness principle, let M = max
Uniform_boundedness_principle
Principle in fluid mechanics
Pascal's rule is applied to confined space (static flow), but due to the continuous flow process, Pascal's principle can be applied to the lift oil mechanism
Pascal's_law
Possibility of a consistent definition of "clockwise" in a mathematical space
starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip
Orientability
Set of points on a line segment with certain topological properties
ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space
Cantor_set
Infinitely detailed mathematical structure
of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable
Fractal
Algebraic structure in linear algebra
approximation of differentiable functions f {\displaystyle f} by polynomials. By the Stone–Weierstrass theorem, every continuous function on [ a , b ] {\displaystyle
Vector_space
Upcoming Marvel Studios television miniseries
renderings in the design of the Mindspace—including stair banisters that lead nowhere, chairs built into walls, and wonky furniture. Vision's pinstriped suit
VisionQuest
Mathematical theorem
only elementary functions. Simply connected open sets in the plane can be highly complicated, for instance, the boundary can be a nowhere-differentiable
Riemann_mapping_theorem
Part of spectral theory
where p is a strictly positive continuously differentiable function and q and r are continuous real-valued functions. For x0 in (a, b), define the Liouville
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Physical quantities taking values at each point in space and time
the mathematical methods of continuous random fields are used, because thermally fluctuating classical fields are nowhere differentiable. Random fields
Field_(physics)
following the third episode of season 6, the show's ratings declined continuously. Another reason was the show's new time slot, which was Saturdays at
List of Petticoat Junction episodes
List_of_Petticoat_Junction_episodes
Property of measure-preserving dynamical systems
weaker statement about averaged translates of square-integrable functions. For a continuous transformation of a second-countable topological space with a
Ergodicity
15th-century codex in an unknown script
Hebrew language can be traced as far back as the ninth century, it is nowhere near as compact or complex as the shapes Newbold made out. Close study
Voynich_manuscript
Branch of pure mathematics
mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical
Number_theory
fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According to Weierstrass in his paper,
List_of_conjectures
Mathematical property of a space
subcover. Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded. σ-compact. A space is σ-compact if
Topological_property
Enjoys playing cruel pranks on fellow Decepticons and appearing out of nowhere to attack Autobots. Not too smart; would be useless without Megatron's
List of The Transformers characters
List_of_The_Transformers_characters
conjectures: every bridgeless graph has a nowhere-zero 5-flow every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow Woodall's conjecture that
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Parameter in differential equations and dynamical systems
the overall system can be evolved in "time", which may be discrete or continuous. The problem of determining a system's evolution from initial conditions
Initial_condition
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NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
NOWHERE CONTINUOUS-FUNCTION
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