# Questions tagged [lipschitz-functions]

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

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### Proving an integral inequality involving Lipschitz functions

I am trying to prove the following inequality: $$\int_a^b \Big( f(a)+f(b)-2f(x) \Big)dx\leq \frac{1}{2}\text{Lip}(f) (b-a)^2$$ where $f$ is a Lipschitz function and $a<b$. This inequality is ...
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### Large-scale Lipschitz and bornologous maps

I'm trying to understand the following questions which I took from Roe's book "Lectures on coarse geometry". A map of metric spaces $f:X \to Y$ is called large-scale Lipschitz if there are ...
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### Show that $\frac{e^x}{1+e^x}$ is lipschitz with Lipschitz-constant $\frac{1}{4}$ [closed]

I am trying to show that $\dfrac{e^x}{1+e^x}$ is lipschitz with constant $\dfrac{1}{4}$. For that I try to show that $\dfrac{e^x}{(1+e^x)^2} \leq \dfrac{1}{4}.$ But I don’t really know how to do this. ...
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### How to intuitively view Wasserstein distance dual as moving earth

The Wasserstein-1 distance can be viewed as the minimum amount of work needed to move one distribution to another distribution, as if the distributions were like piles of earth. The typical definition ...
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I read a statement in a paper and I do not think it is correct. Let $X = L^2(\Omega) \times L^2(\Omega) \times L^2(\Omega)$ where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain. For $u = (f,... 6 votes 3 answers 172 views ### Lipschitz constant tending to zero implies function is constant Let$f$be a Lipschitz function defined on a ball$B$in$\mathbb{R}^n, and assume that the following property on the Lipschitz constant (restricted to smaller balls) holds: $$\lim_{r\to 0}\sup_{|x-... 1 vote 1 answer 36 views ### Is the function f(x, y)= x\sin y+y\cos x Lipschitz on [-a, a]\times[-b,b]? My attempt Let (x, y_1) and (x, y_2) be two points in the rectangle R=[-a, a]\times[-b,b]. Then$$|f(x, y_1)- f(x, y_2)|\le|x||\sin y_1 -\sin y_2| + |y_1-y_2|$$I can't understand how to proceed ... 2 votes 1 answer 22 views ### Separation of normalized molecules in Lipschitz free space For a pointed metric space (M,d,0) (meaning (M,d) is a metric space and 0\in M is a distinguished point), \text{Lip}_0(M) is the Banach space of all real-valued Lipschitz functions f:M\to \... 3 votes 1 answer 60 views ### Is the tangent space of a Riemannian manifold a local approximation of the manifold? Let (M,g) be a Riemannian manifold, p \in M. You will often hear people say "(T_pM, g(p)) is the infinitesimal geometry of M at p". I would like to upgrade `infinitesimal' to '... 2 votes 1 answer 61 views ### Improve the bound of a Lipschitz function Let \lambda be a function defined as$$\lambda(x) = \frac{x}{|\ln (x)|^2}$$on an interval [0,x_0] for some x_0 \ll 1 (x_0 can be chose arbitrarily small). Now let us consider \varepsilon ... 0 votes 1 answer 52 views ### Determining when the function f(x)=\int_{0}^x g is Lipschitz and finding the best Lipschitz constant when it is. Let E=(0, \infty) and g \in L^{1}(E). Define f\colon E \to \mathbb{R} by$$f(x)=\int_{0}^{x}g.(a) Show that f may not be Lipschitz on E. (b) Prove that if g \in L^{\infty}(E) then f ... 0 votes 1 answer 26 views ### Show the error squared of the Euclidian projection onto a closed set is a continuous function I asked this and now I am wondering if the previous proof can be used to show the following: Let f(x)=\min_{y \in C}\|y-x\|^2 where C is a closed set in \mathbb{R}^n and x,y \in \mathbb{R}^n. ... 2 votes 0 answers 24 views ### Hausdorff measure of f(E) is greater than zero where E compact and f is (...)? I'm studying example 4.1.3 of "Variational Analysis in Sobolev Spaces and BV Spaces" by Attouch, Buttazzo and Michaille. At one point it states that Given f: R^n \to R^m \> (n\leq m) ... 0 votes 0 answers 59 views ### Length distance and distance generated by a set of functions. Let X be a non-empty set and let A be an algebra of bounded functions on X containing the constant functions and separating the points in X. Given a distance \rho on X, we define \rho_\... 0 votes 0 answers 266 views ### Piecewise defined Lipschitz function Suppose B \subset \overline{\mathbb{C}} is an open set not containing the point at infinity, v \in C(\overline{B}) is valued in [-1,1], and Lipschitz on compact subsets of B and \omega is ... 1 vote 0 answers 39 views ### Growth assumptions on nonlinear term in PDE In Partial Differential Equations book by Evans they treat a nonlinear system of reaction-diffusion equations. The nonlinearity comes from the reaction term f \begin{align*} & \partial_t u - \... 1 vote 1 answer 46 views ### Can I find a lipschitz constant for this term? I have given two functions f(x)=\frac{1}{2}x and g(x)=\sqrt{1+x^2}, x\in \Bbb{R}, I want to show that for some K\geq 0,|f(x)-f(y)|+|g(x)-g(y)|\leq K|x-y|$$for all x,y\in \Bbb{R}. And ... 0 votes 0 answers 26 views ### Calculating Lipschitz constant of an unknown function from its data I have an unknown four-dimensional function:$$u:X \subset \mathbb{R}^4 \to Y\, \text{where}\, X := [-5,5] \times [-5,5]\times [-5,5]\times[-5,5],\,Y= [-0.2,0.2],$$and I have access to its values on ... 2 votes 0 answers 49 views ### Looking for regularity classes which are weaker than continuity To contextualize my question, recall that Lipschitz (i.e. C^{0,1}) functions by Rademacher's theorem are differentiable almost everywhere. It's interesting to note that this is sharp (and apparently ... 1 vote 1 answer 32 views ### Check proof that |f_{a_1}(x)-f_{a_2}(x)| \leq f'(x) |a_1 - a_2| Let a_1, a_2 \in A where A is a compact set. Let x \rightarrow f_a(x) be a measurable function such that a \rightarrow f_a(x) is differentiable for almost every x. I have to prove that there ... 1 vote 1 answer 56 views ### Class of Lipschitz Functions on the unit d-dimensional ball Let \mathcal{F} = \{f:\mathcal{B}_d \to \mathbb{R}\;:\; \text{f is Lipschitz}\}, where \mathcal{B}_d = \{x \in \mathbb{R}^d\;:\: \|x\|_2 \leq 1\} is the unit ball in d dimension. Is the class \... 4 votes 0 answers 569 views ### Lipschitz functions (Theorem 1.4 of Condenser Capacities and Symmetrization in Geometric Function Theory) I am struggling to understand the second part of the following proof of Theorem 1.4 of the book "Condenser Capacities and Symmetrization in Geometric Function Theory" by Vladimir N. Dubinin (... 0 votes 0 answers 67 views ### Sufficient conditions for f\cdot f^\prime to be lipschitz-continuous! Let a>0 (even sufficiently small) and f:[-a,a]\to\mathbb{R} be a Lipschitz-continuous function such that f^\prime is \alpha-Hölder continuous with \alpha<1 (so not lipschitzian). ... 0 votes 1 answer 17 views ### K-Lipschitz surjection to higher dimension I am looking for a (family of) surjective K-Lipschitz function f : [0,1]^n \rightarrow [0,1]^m where n < m. I am more interested in an explicit description of such a function than in a proof of ... 0 votes 1 answer 48 views ### A Lipschitz continuity-style problem [duplicate] f is a function defined on an interval I. \exists K\:>0 such that \forall x,y\in I, \left|f\left(x\right)-f\left(y\right)\right|\le K\left|x-y\right|^2. Show that f is constant on I. f ... 5 votes 0 answers 171 views ### Existence and uniqueness of the solution of a control system Let T>0, (U,d) be a metric space and \mathcal{V}:=\{u:[0,T]\to U\,|\, u\text{ is measurable}\}. Consider the control system$$\begin{cases}\dot{x}(t)=f(t,x(t),u(t)),\quad \text{a.e. }t\in[0,T]... 1 vote 1 answer 54 views ### Prove that iff$satisfies the Lipschitz condition, then the solutions to$\frac{dx}{dt} = f(x)$are defined for all$t \in \mathbb R$Prove that if$f : \mathbb R^n \to \mathbb R^n$satisfies the Lipschitz condition, then the solutions to$\frac{dx}{dt} = f(x)$are defined for all$t \in \mathbb R$. Suppose the function$f$... 0 votes 1 answer 48 views ### Conditions for the squared gradient norm to be convex? Let$f\colon \mathbb{R}^n \to \mathbb{R}$be a differentiable function. I am looking for conditions under which the function $$x\mapsto \Vert\nabla f(x)\Vert^2$$ is convex. It obviously hold if$f$is ... 0 votes 1 answer 45 views ### Inequality for Subgaussian Norms and Lipschitz Functions If I have a$L$-Lipschitz function$\psi \colon \mathbb{R} \longrightarrow \mathbb{R}$with$\psi(0) = 0$, and a random variable$X \sim \mathcal{N}(0, \sigma^2), is it true that $$\|\psi(X)\|^2_{\... 1 vote 1 answer 67 views ### The inverse of an arc length parametrization of \partial U is Lipschitz Let U be a simply connected open bounded subset of \mathbb{C}\cong\mathbb{R}^{2} with smooth boundary (thus the boundary \partial U is diffeomorphic to a circle). Let L\geq0 denote the total ... 3 votes 2 answers 68 views ### Is a holomorphic f\colon U\to\mathbb{C} with continuous extension to \overline{U} Lipschitz continuous on \partial U? Let U\subset\mathbb{C} be a bounded connected open subset with smooth boundary \partial U. Suppose that we have a holomorphic function f\colon U\to\mathbb{C} that can be continuously extended to ... 1 vote 0 answers 48 views ### Lipschitz Continuity and Lipschitz Constant of a Multidimensional Function I have a function f: \mathbb{R}^n \to \mathbb{R} defined as follows:$$ f(x_1,\dots,x_n) = -\frac{x_j-c_j}{(1-k_j)^2}e^{ -\frac{1}{2}\sum_{i=1}^n\left(\frac{x_i-c_i}{1-k_i}\right)^2} for j \in \{... 0 votes 0 answers 27 views ### Lipschitz Hessian implies Lipschitz Hessian diagonal for non-convex function? I am working on an optimization problem where the function f is assumed to have Lipschitz-continuous gradients and Hessian \begin{align} \| \nabla^2 f(x) - \nabla^2f(y) \| \leq L_1 \| x -y \|, \... 0 votes 0 answers 29 views ### Real analysis, Uniform boundedness and Uniform convergence Suppose that f_{n}be a sequence of real-valued functions that are uniformly Lipschitz and uniformly bounded on \left[0,1\right],there exists constants K,M>0 such that for all n\geq 1 one ... 1 vote 1 answer 66 views ### Lipschitz continuous compactly supported function in polar coordinates Let f be a lipschitz continuous complex valued function in the complex plane, compactly supported in the closed disk of radius \delta > 0 with lipschitz constant M. Define h(r,\theta) = f(r \... 0 votes 1 answer 44 views ### Proving that a function G: X\rightarrow \mathbb{R}^{n+1} is continuous. (X, ||\cdot||) is a normed vector space, and l_0, l_1, ..., l_n are linear operators in X^*. And G is defined like this. G:X\rightarrow R^{n+1}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x)). ... 0 votes 2 answers 63 views ### Lipschitz continuous function | Real Analysis Let a, \space b \in \mathbb{R} with a < b and consider f:[a,b] \rightarrow \mathbb{R} s.t f is continuous at [a, b] and Lipschitz continuous at (a, b). Does it necessarily imply f ... 3 votes 2 answers 70 views ### Show that in gradient descent the gradient goes against 0 (Under certain conditions) So we have gradient descent:x^{(i+1)} = x^{(i)} - \tau \nabla f(x^{(i)})$$And we gotta show that$$\left|\nabla f\left(x^{(j)}\right)\right| \to 0$The conditions are:$f: \mathbb R^n \to \...
We know by Rademacher's theorem that a Lipschitz-continuous map is differentiable almost everywhere (a.e.). Now for every path in $\mathbb{R}^n$, $n>1$, there is a Lipschitz-continuous map such ...
Let $M$ be a closed Riemannian manifold (perhaps even a surface, at first), and let $u: M \to \mathbf S^1$ be a Lipschitz map to the circle, with Lipschitz constant $L$. We can find an approximation \$(...