Articles of lipschitz functions

Lipschitz continuity implies differentiability almost everywhere.

I am running into some troubles with Lipschitz continuous functions. Suppose I have some one-dimensional Lipschitz continuous function $f : \mathbb{R} \to \mathbb{R}$. How do I prove that its derivative exists almost everywhere, with respect to the Lebesgue measure? I found on other places on the internet that any Lipschitz continuous function is absolutely continuous, […]

The function $4x^3y/(x^4+y^2)$ fails the Lipschitz condition near the origin

I have to prove that Lipschitz condition is not satisfied for the function, $$ f(x) = \begin{cases} {4x^3y \over x^4 +y^2}, & \text{if $(x,y) \neq (0,0)$ } \\ 0, & \text{if $(x,y)=(0,0)$ } \end{cases}$$ throughout any domain which includes $(0,0)$. I considered ,the domain $D = \{(x,y) : |x| \le a , |y|\le b,a \gt […]

(dis)prove:$\sup_{F \in 2^{(L^1(S,\mathbb{R}))}}\limsup\sup_{f\in F}|\int f dP_n-\int fdP|=\limsup\sup_{f\in L^1(S,\mathbb{R})}|\int fdP_n-\int fdP|$

Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) – f(y)| \leq d(x,y)$. Further let: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid P \mbox{ probability measure:} \forall a\in S: \int d(a,x) P(dx) < \infty \}, $$ […]

The origin is not in the convex hull $\Rightarrow$ the set lies in a hemisphere?

I am trying to understand the proof of the following claim: Let $f:A \subseteq \mathbb{S}^n \to \mathbb{S}^n$ be an $L$-Lipschitz* map (with $L <1$). Then $f(A)$ is contained in the interior of a hemisphere. *The distance on $\mathbb{S}^n$ can be either the intrinsic one or the extrinsic (Euclidean) one, it does not matter. In the […]

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i’m not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, and let $$ BL(\mathcal{X})=\{f:\mathcal{X}\to \mathbb{R}\, \, | \, \, f \text{ is Lipschitz and bounded}\} $$ denote the bounded real-valued Lipschitz […]

Prove $\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alpha}\int_c^d f(x)dx$

Assume $f(x)>0$ defined in $[a,b]$, and for a certain $L>0$, $f(x)$ satisfies the Lipschitz condition $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$. Assume that for $a\leq c\leq d\leq b$,$$\int_c^d \frac{1}{f(x)}dx=\alpha,\int_a^b\frac{1}{f(x)}dx=\beta$$Try to prove$$\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alpha}\int_c^d f(x)dx$$

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz’s condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an example to prove it sufficient. That is I want an example of an I.V.P. of the form $$\frac{dy}{dx}=f(x,y)\text{ , with initial condition […]

Is a function Lipschitz if and only if its derivative is bounded?

Is the following statement true? Let $f: \mathbb{R}\to\mathbb{R}$ be continuous and differentiable. $f$ Lipschitz $\leftrightarrow \exists M:\forall x\in\mathbb{R}\ |f'(x)|\leq M$ If $f’$ is bounded, it is Lipschitz, that’s obvious. Does that work the other way around? Let $f$ be $M$-Lipschitz, that is to say $\forall x_1, x_2\in\mathbb{R},\ |f(x_1) – f(x_2)| \leq M|x_1 – x_2|$, where […]

Lipschitz $\implies |f'(x)|\le d$

Let $U\subset \mathbb R^m$ be a convex subset and $f:U\to \mathbb R^n$ be a differentiable function and I would like to prove the following equivalence: $$|f'(x)|\le d\ \text{for every $x\in U$}\Leftrightarrow |f(x)-f(y)|\le d|x-y|\ \text{for every $x, y\in U$}.$$ The $\Rightarrow$ is easy I used the Mean Value Theorem with the fact $U$ is convex. I’m […]

Show continuity or uniform continuity of $\phi: (C(;\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $ Is this function continuous or even uniformly continuous? (I know that the function $g: M\to \Bbb R , g(x) := x^2$ is continuous but not uniformly) Also, is there a non-empty subset of $ C([0,1];\Bbb […]