of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such...

can only have jump discontinuities; f {\displaystyle f} can only have countably many discontinuities in its domain. The discontinuities, however, do not...

considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem...

of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence...

continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) F : R → [ 0 , 1 ] {\displaystyle F\colon \mathbb {R}...

Dirichlet in 1829, for piecewise monotone functions (functions with a finite number of sections per period each of which is monotonic). It was extended...

half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the...

mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use:...

example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous...

this puts the validity of the regression discontinuity design into question. If discontinuities are present at other points of the assignment variable...

sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly...

{y}}\neq y} , and 0 otherwise. In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation...

bounded monotone. In particular, a BV function may have discontinuities, but at most countably many. In the case of several variables, a function f defined...

real-valued function is monotone on the interval [a, b] it is Riemann integrable, since its set of discontinuities is at most countable, and therefore of Lebesgue...

continuous. A function g is of bounded variation if and only if it is the difference between two (bounded) monotone functions. If g is not of bounded variation...

to the theory of maximal monotone operators, as developed by Minty and Haïm Brezis. Filippov's theory only allows for discontinuities in the derivative...

spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate...

\langle L,\leq \rangle } and a monotone function f : L → L {\displaystyle f\colon L\rightarrow L} on L, the set of all fixpoints of f is also a complete lattice...

advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing...

monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities...

distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two...

monotonically non-increasing functions of t , {\displaystyle t,} so both of them have at most countably many discontinuities and thus they are continuous...

variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value...

Harten 1983 proved the following properties for a numerical scheme, A monotone scheme is TVD, and A TVD scheme is monotonicity preserving. In Computational...

The interpolation functions, either polynomials or trigonomic functions are global in nature. Their main benefits is in the rate of convergence which...

regime, the non-monotone evolution of the Mach number in isentropic expansions can be found even in subsonic conditions. In fact, for values of Γ < 0 {\displaystyle...

{\displaystyle \Gamma ,K\in \mathrm {Trans_{P}} } (a function from P {\textstyle P} to P {\textstyle P} which is monotone and satisfies x ≤ Γ ( x ) {\displaystyle...

semimartingale. The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process...

recognized domains of negative symptoms are: blunted affect – showing flat expressions (monotone) or little emotion; alogia – a poverty of speech; anhedonia...

(mathematics) Moment-generating function Moments, method of – see method of moments (statistics) Moment problem Monotone likelihood ratio Monte Carlo integration...