# 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 $y$.

927 questions
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### Yosida Transform's pointwise convergence.

Let $f = \Gamma-\lim_{j\to\infty} f_{j}(x)$, where $f_{j}:\mathbb{R}\rightarrow [0, +\infty]$ are convex and lower semicontinuous functions such that $\sup_{j} f_{j}(0)< \infty$. I have to prove ...
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### Closedness of Bi-Lipschitz embedding's image

Suppose that $(X,d)$ and $(Y,\rho)$ are complete and separable metric spaces. If $\phi:X\rightarrow Y$ is a bi-Lipschitz embedding, is it the case that $\phi(X)$ is closed in $(Y,\rho)$?
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### Is the metric space of Lipschitz Function with $d_{\infty}$ complete?

This is my real analysis homework. Define $$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$ and the metric $$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$ ...
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### Is entropy function lipschitz? [on hold]

Is the information entropy function Lipschitz with respect to $l_{1}$ norm or $l_{2}$ norm?
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### How to show dual norm of subgradient of a $L$-Lipschitz convex function is bounded by $L$?

I am studying the monograph Online Learning and Online Convex Optimization. At page 133, the author has the following Lemma: $\textbf{Lemma 2.6.}$ Let $f: S \rightarrow \mathbb{R}$ be a convex ...
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### What is the intuition behind having upperbound on eigenvalues of Hessian?

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and is a $C^1$, i.e., $\nabla f$ is a continuous vector valued function. Show if the $\lambda_{max}$ of $\nabla^2f$ is bounded, then $\nabla f$ is ...
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### Proving Lipschitz Continuity

Let $X$ be a Hilbert Space, $\varphi \in C^{2}(X,\mathbb{R})$, $c\in\mathbb{R}$, and $\varepsilon > 0$. Denote $\varphi^{-1}(\,\cdot\,)$ as the pre-image of $\varphi$ and define \begin{align*} A &...
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### Equivalence of statements regarding convex function with Lipschitz continuous gradient

I came into a practice problem on Lipschitz continuous gradient. Given convex and twice continuously differentiable $f$, prove the following statements are equivalent. $\nabla f$ is ...
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### Does the boundary of a level set of a Lipschitz continuous function have Lebesgue measure $0$?

Let $f:\mathbb R\to\mathbb R$ be a (bounded, if necessary) Lipschitz continuous function. Are we able to show that $\partial f^{-1}\left(\left\{0\right\}\right)$ has Lebesgue measure $0$. If not, are ...
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### Lipschitz condition on $x*|y|$

The IVP $y'=x|y|$ is given along with the condition $y(1)=0$. Upon checking the Lipschitz condition one gets, $|x|*||y_{2}|-|y_{1}||\le|x|*|y_{2}-y_{1}|$ Now, if I go locally around $x=1$, $|x|$ ...
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