0
votes
0answers
56 views

Continuity of the real Monge Ampère operator on convex functions

Let $E\subset\mathbb{R}^n$ be a convex set, $u:E\to\mathbb{R}$ a convex function and $B\subset E$ a Borel set. We define the (multivalued) gradient \begin{array}{rccl} \nabla [u]:& \mathbb{R}^n ...
5
votes
3answers
635 views

On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$. I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...
2
votes
1answer
123 views

Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book "Understanding Analysis" by Abbott is stated in page 128 ...
5
votes
0answers
197 views

Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous. Question: Under what ...