Arbitrary Mixed Partial Derivatives
Background: Many textbooks give an example due to Peano of the function $f(x, y) = xy(x^2 - y^2)(x^2 + y^2)^{-1}$ that has mixed partial derivatives at $0$ that are not symmetric. One might wonder how...
View ArticleNorm of the triangular $SL(2)$-matrix
Let $A=\begin{pmatrix}4 & 0 \\1 & \frac{1}{4}\end{pmatrix}$. What is the supremum norm $\|A^2\|$? Recall that $\|A\|:=\sup\{\frac{|Av|}{|v|}: v\neq 0\}$.Attempt: $A^2=\begin{pmatrix}4^2 & 0...
View ArticleDifferentiation on an arbitrary set
The subject of this question is differentiability of a function on an arbitrary set.It doesn't matter if we exclude isolated points from the domain of a function. So, let $X$ be a subset of...
View ArticleA continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either...
I would like to prove the statement in the title.Proof: We prove that if $f$ is not strictly decreasing, then it must be strictly increasing.So suppose $x < y$.And that's pretty much how far I got....
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Inequality involving entropies: $\left \|p -\frac{1}{n} e \right \|_2\ge\left...
For a given probability vector $p=(p_1,\dots,p_n)$ with $p_1,\dots,p_n > 0, \sum_{i=1}^n p_i=1$ and with $e:= (1, \dots, 1)$, I want to prove the following inequality:$$\small\left \|p -\frac{1}{n}...
View ArticleShow that $\left \lVert \int_0^1 F(t) dt \right \rVert \leq \lVert F...
Let $F:[0,1] \to \mathbb{R}^n$ be a continuous function such that $F=(f_1,...,f_n)$ and let:$$\int_0^1 F(t) dt := \left( \int_0^1 f_1(t) dt,...,\int_0^1 f_n(t) dt\right)$$And:$$\lVert F \rVert_{\infty}...
View ArticleRiemann-Stieltjes Integral Definition
I have been taking a university course in which they define the Riemann-Stieltjes Integral using upper sums and lower sums of a function and then we take their respective infimum and supremum.In the...
View ArticleDoes it follow that $\mu$ is a measure? [duplicate]
Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$....
View ArticleSquashing/zeroing higher order derivatives of a function at a point [closed]
I have certain rational functions (both numerators and denominators are polynomial types) but I want certain higher order derivatives at some point $p_0$ to be zero, without affecting much the behavior...
View ArticleDoes application of epsilon delta definition for limits of a real function...
I believe it doesn't but it could be I am wrong.If we have a sequence $ \{\frac{1}{n}\}_n$ then the archimedian property plays a central role in proving the limit. For instance, let $\epsilon$ be an...
View ArticleDoes this proof about an inequality related to fourier transforms work?
Question:If $f$ is real-valued and continuously differentiable on $\mathbb{R}$, prove that$\left(\int |f|^2 dx\right)^2 \leq 4 \left(\int |xf(x)|^2 dx\right) \left(\int |f'|^2 dx\right).$Attemped...
View ArticleShow $V_{F_{\mu}}(-\infty,x]=|\mu|((-\infty,x])$ for all $x\in\mathbb{R}$...
My QuestionI want to prove the following:Claim$\quad$Let $\mu$ be a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Show that $V_{F_{\mu}}(-\infty,x]=|\mu|((-\infty,x])$ for all...
View ArticleDoes there exist a continous function for which there is no continuous choice...
We say that a function has the property of having continuous choice of $\delta$ provided $\epsilon$ for each point in it's domain, if there is an assignment of $\delta$ for each provided $\epsilon$...
View ArticleProof: The Variation of $V_{F_{\mu}}$ is finite.
Background InformationSuppose that $\mu$ is a finite signed measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Define a function $F_{\mu}:\mathbb{R}\to\mathbb{R}$ by letting$$F_{\mu}(x) =...
View ArticleWill a bounded sequence in $\mathbb{R}$ necessarily induce a compact subset...
Question: We know that a bounded sequence in $\mathbb{R}$ must have a convergent sub-sequence from Bolzano-Weierstrass theorem. However, will a bounded sequence $(x_n)$ in $\mathbb{R}$ necessarily...
View ArticleDoes this derivation of the sum formula for geometric series make sense?
I want to prove that $\sum_{k=0}^\infty ar^k = \frac{a}{1-r}$ when $|r| < 1$. If I can prove the sequence of partial sums $S_n$ converges to said formula, then I will have proven the series does as...
View ArticleUniform semi-continuity
BackgroundIt is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose $(X,d)$ is a compact...
View ArticleUniformly coverging sequence of functions and differentiablity
Let $f_n:\mathbb R \rightarrow \mathbb R$ be a sequence of functions such thateach $f_n$ can be extended to an entire function $g_n: \mathbb C \rightarrow \mathbb C$.Let$$\Omega = \{ x + y i \in...
View ArticleLet a be some positive real number. Consider the function f : (0,a) -> S...
Let a be some positive real number. Consider the function f : (0,a) -> S defined as f(t) = exp (2pie i t). Identify all the values of a for which f a surjective local homeomorphism, but not a...
View ArticleGiven a closed set in a preordered metric space. Can we found its minimum value?
Let $(X,d)$ be a metric space, and also $(X,\le)$ is a preordered set. If given a subset $E\subset X$ and $E$ is a closed set, can we always find the minimum value $\beta$ of $E$, such that $\forall...
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