Is it true that $\frac{b-a}{a}\geq\ln\frac{b}{a}\geq\frac{b-a}{b}$ for $0
By the mean value theorem, for $0<a<b$, one has$$\frac{b-a}{b}\leq\ln\frac{b}{a}\leq\frac{b-a}{a}.$$Does this imply that, for $0<b<a$, one...
View ArticleConvexity of a function defined as the square of a maximum of functions
\textbf{5.} Analyze whether the function ( f(x) = (g(x))^2 ) is a convex function, where[g(x) = \max{2 + x, , x, , x^2 - 4}.]
View ArticleEvans PDE prerequisite
In Appendix C of Evans's PDE textbook, lots of results about the boundary of open and bounded subsets of $\mathbb{R}^n$ and surface integrals against such boundaries are mentioned. Is there any...
View ArticleConvexity of $f(x) = g^2(x)$, where $g(x) = \max\{2 + x, x, x^2 - 4\}$ [closed]
How can we prove the convexity (convex or concave) of the function $$f(x) = g^2(x),$$ where $g(x) = \max\{2 + x, x, x^2 - 4\}$?Any hints will be highly appreciated!
View Articleproving $ f\left( x\right)= x $ is riemann integrable over $ \left[...
I want to prove that $ f\left( x\right)= x $ is riemann integrable over $ \left[ 0,1\right] $ using the Squeeze Theorem here is my approach i want to know if it is right ? let $\epsilon $ be given Let...
View ArticleProving that if $(X,d)$ is a complete and infinitely countable metric space,...
I tried proving it as follows:Suppose for a contradiction that all $x \in X$ are not open, since the metric space is countable, this means $X = \cup_n {x_n}$. Because every singleton is not open and...
View ArticleSimplest proof of Taylor's theorem
I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem.By which I mean this: there are plenty of proofs that introduce some arbitrary construct: no...
View ArticleAtomless probability space and Random Variable with continuous distribution
Let $(\Omega, \mathcal{F}, P)$ a probability space. We say that a set $A\in\mathcal{F}$ is an atom if $P(A) > 0$ and for all $B \in \mathcal{F}$ such that $B \subset A$ if $P(B) < P(A)$ implies...
View ArticleUnicity of solution of ODE when the function is continuos and non-decreasing...
I'm studying Ordinary Differential Equations using the book of Viana and Spinar, and I came a cross with a question that I can't formalize (Ex 2.12):Let $F: U \subset \mathbb{R}^2 \rightarrow...
View ArticleSummation of two equicontinuous family of functions
Let $X$ be a metric space. Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of real valued functions. If $\mathcal{F}$ and $\mathcal{G}$ are equicontinuous, is $\mathcal{F}+\mathcal{G}$...
View ArticleUnion of two equicontinuous families of functions is equicontinuous
Let $X$ be a metric space, $\mathcal{F}$ and $\mathcal{G}$ are families of real valued functions. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are equicontinuous. I ever read that union of two...
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 ArticleShow that if $F_{1}(x_{1}) \cdots F_{n}(x_{n}) = F(x_{1}, \ldots, x_{n})$,...
Let $X_{1}, \ldots, X_{n}$ be random variables. FOr each $k \in \{1, \ldots, n\}$, the cumulative distribution function for $X_{k}$ is the function $F_{k}$ defined on $\mathbb{R}$ by $F_{k}(x) =...
View ArticleSuppose $ f(0) = 0 $ and $ |f'(x)| \leq |f(x)| $ for every $ x \in (0,1) $....
Let $ f $ be continuous on $ [0,1] $ and differentiable on $ (0,1) $. Suppose $ f(0) = 0 $ and $ |f'(x)| \leq |f(x)| $ for every $ x \in (0,1) $. Show that $ f(x) = 0 $ for all $ x \in [0,1] $.This...
View ArticleShow that $-2
Given $f(x)=\frac{1-x}{e^{x-2}}$, prove that: $$-2<\ln x \cdot \int_{0}^{x} f(t) {\rm{d}}t<\sqrt{x}$$It is easy for me to show $\int_{0}^{x} f(t) {\rm{d}}t = x {\rm{e}}^{2-x}$, but then I have no...
View ArticleConvex Real Function is Pointwise Supremum of Affine Functions
I'm read the proof of the Convex Real Function is Pointwise Supremum of Affine Functionshttps://proofwiki.org/wiki/Convex_Real_Function_is_Pointwise_Supremum_of_Affine_FunctionsBut I dont understand...
View ArticleIntersections of intervals bounded by suprema and infima.
Let $(a_k)_{k=m}, (b_k)_{k=m}$ be sequences of real numbers. If $m\le p$ define $\alpha_p$ to be the infimum of the $p$th tail, and similarly for...
View ArticleWhere is the error in my proof of the quotient rule for $(f/g)'$?
So we all know that, if $f$ and $g$ are defined on $[a,b]$ and are differentiable at a point $x \in [a,b]$, then $f/g$ is differentiable at $x$ if $g(x) \neq 0$. And the quotient rule says that $\left...
View ArticleThe Geometry of Fractal Sets by K. J. Falconer - Exercise 1.8.
I'm stuck with the following exercise from the book The Geometry of Fractal Sets by K.J. Falconer:Let $\mu$ be a Borel measure on $\mathbb{R}^n$ and let $E$ be a $\mu$-measurable set with $0 <...
View ArticleIf $f$ is a continuous mapping of metric spaces $X$ into $Y$, prove that...
If $f$ is a continious mapping of a metric space $X$ into a metric space $Y$, prove that $f(\overline{E}) \subset \overline{f(E)}$ for all $ E \subset X$($\overline{E}$ denotes the closure of $E$)This...
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