When does restricted continuity on continuous function graphs imply joint continuity?

Continuity of super- and sub-additive transformations of continuous functions

We prove a continuity inheritance property for super- and sub-additive transformations of non-negative continuous multivariate functions defined on the domain of all non-negative points and vanishing at the origin. As a corollary of this result we obtain that super- and sub-additive transformations of continuous aggregation functions are again continuous aggregation functions.

Contents 0 Foreword 3 1 Function Limits and Continuity 3 2 Continuity 4 3 Continuous Functions on Closed Intervals 5

Density Continuity Versus Continuity

Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if...

Real Function Continuity 1 Konrad

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...

Oscillations, Quasi-oscillations and Joint Continuity

Parallel to the concept of quasi-separate continuity, we define a notion for quasi-oscillation of a mapping f : X × Y → R. We also introduce a topological game on X to approximate the oscillation of f . It follows that under suitable conditions, every quasi-separately continuous mapping f : X×Y → R has the Namioka property. An illuminating example is also given.