# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,484 questions
110 views

### Given these conditions, can a function be defined that is well defined a.e.?

I have two functions, and I want to combine them to define a certain function. Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a....
41 views

113 views

### Kantorovich duality with pseudometrics

The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that ...
36 views

### Potential for a Monotone Operator

[Cross-posted from math.stackexchange] I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
190 views

### Oscillation operator of a function

Call a function from $[0, 1]$ to itself a box function. Given any box function $f$, define its oscillation function $Of$ as $$Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)| \, .$$ ...
83 views

### Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
93 views

44 views

### Does lattice mod preserve direction?

For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...
54 views

### Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$. Which vector fields $F$ are so that ${\rm div} (F)= f$ ?
123 views

### $L^1$-continuity estimate for ODE solutions in terms of $L^1$ distance of vector fields (only one of them being Lipschitz)

Consider the following ODE initial value problems \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in ...
666 views

### “Insanely increasing” $C^\infty$ function with upper bound

Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...
131 views

406 views

218 views

41 views