Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric $M$ that vanish at base point. We show every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering question by N. Weaver.

The continuity of real functions is discussed. There is a function defined on some domain in real numbers which is continuous in a single point and on a subset of domain of the function. Main properties of real continuous functions are proved. Among them there is the Weierstraß Theorem. Algebraic features for real continuous functions are shown. Lipschitzian functions are introduced. The Lipsch...

0.1 Model prerequisites Consider ˙ x = f (t, x). We will make the following basic assumptions ensuring that this model can be used for evolving the state x: f (t, x) is piecewise continuous in t and locally Lipschitz, i.e.: • f (t, x) is piecewise continuous if f is continuous on any subinterval of t except at, possibly, finite number of points where it might have finite-jump discontinuitiies

This note presents an implicit function theorem for generalized equations, simultaneously generalizing Robinson’s implicit function theorem for strongly regular generalized equations and Clarke’s implicit function theorem for equations with Lipschitz-continuous mappings.

Let g be a locally Lipschitz continuous real valued function which satisfies the KellerOsserman condition and is convex at infinity, then any large solution of −∆u+ g(u) = 0 in a ball is radially symmetric. 1991 Mathematics Subject Classification. 35J60 .

We prove existence and uniqueness of integral curves to the (discontinuous) vector field that results when a Lipschitz continuous vector field on a Hilbert space of any dimension is projected on a non-empty, closed and convex subset.

It is shown that there exist Banach spaces X,Y , a 1-net N of X and a Lipschitz function f : N → Y such that every F : X → Y that extends f is not uniformly continuous.

We analyze the reducibilities induced by, respectively, uniformly continuous, Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces, and determine whether under suitable set-theoretical assumptions the induced degree-structures are well-behaved.

The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for class asymptotically stable infinite dimensional systems with unbounded nonlinear operators.