$f$ strictly increasing does not imply $f'>0$
We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'>0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any...
View ArticleHow $x^4$ is strictly convex function?
I hear that $f(x)=x^4$ is a strictly convex function $\forall x \in \Re$. However, strict convexity condition is that the second derivative should be positive $\forall x \in \Re$.For the mentioned...
View ArticleUnderstanding a proof that a set equals to its closure implies the set is...
This proof is presented in V.A. Zorich Mathematical Analysis I p.415. I have done some research on the internet but it seems like both the definitions and the proof used in this book are unique.The...
View ArticleStein's Real Analysis Chapter 4 Exercise 16
I'm having a trouble with the exercise 16 in chapter 4 of Stein's "Real Analysis, Measure Theory, Integration, Hilbert spaces".Let $F_0(z) = 1/(1-z)^i.$(a) Verify that $|F_0(z)|\le e^{\pi/2}$ in the...
View ArticleLooking for a functional satisfying $f(xf(y)) = \frac{f(x)}{y}$
I am looking for a functional $f:\mathbb{R}^*_+ \to \mathbb{R}^*_+$ satisfying:$$f(xf(y)) = \frac{f(x)}{y}.$$By replacing $x=y=1$ we get $f(f(1)) = f(1)$ but it does not give much information so $f...
View ArticleQuestions about the extended real number system: Is...
Context: I am currently reading Rudin's real analysis and in chapter $1$ definition $1.23$ the extended real number system $\mathbb{\bar{R}}=\mathbb{R} \cup \{ -\infty , \infty \} $ was introduced, his...
View ArticleDecreasing exponential
Consider the function $$f(x) = A e^{xB} e^{-Ce^x} (1 + O(e^{-x})),$$ where $ A, B, C \in \mathbb{R} $, under what conditions on $A, B,$ and $C$ will $f(x)$ be decreasing as $x$ goes large? How do the...
View ArticleProof: Sum of minimum of two functions vs minimum of sum of functions
I wonder whether the following is true:$$\min\big\{f_1(x),g_1(x)\big\} + \min\big\{f_2(x),g_2(x)\big\}\le \min\big\{f_1(x)+f_2(x),~g_1(x) + g_2(x)\big\}.$$I already know how to prove...
View ArticleDetermine whether the following infinite sum converges or not.
Let $\lambda, \mu \in \left (0, \frac {1} {2} \right ).$ Then determine whether the following sum converges or not $:$$$\sum\limits_{n = 0}^{\infty} \left \lvert \sum\limits_{m = 0}^{n} \frac {\Gamma...
View ArticleUsing the derivative, find $a$ such that: $|x-a| = (x-2)^2$
So I've been studying Jay Cummings's Real Analysis book, and I've encountered this problem:Use the derivative to find all values of $ a $ such that the following holds: $$|x-a| = (x-2)^2 $$He doesn't...
View ArticlePercentage increase and decrease of two variables
We have $2$ variables $x$ and $y$ both of which are percentages. If both $x%$ and $y%$ decrease over time (from let's say $t_0$ to $t_1$), can $(x+y)%$ increase in the same time window from $t_0$ to...
View ArticleTest if the series converge $\sum_{n=1}^{\infty} \frac{1}{\sqrt n}$ using...
$$\sum_{n=1}^{\infty} \frac{1}{\sqrt n}$$Obviously, this is the p-series (hyperharmonic series) with $p=\frac{1}{2} < 1$ which means that it is divergent, so that's what I'm going to try to prove...
View Article$\mathbb E(XY) \ge0 \Rightarrow \mathbb E(X \log Y) \ge0$
When does the following implication hold:$$\mathbb E(XY) \ge0 \Rightarrow \mathbb E(X \log Y) \ge0 \tag{1}$$where $\mathbb E(X)=0$, and $Y$ is a non-negative random variable with $\mathbb E(Y)=1? $This...
View ArticleShow $f(f(x))-x$ is monotonically decreasing on $(\frac 1{\sqrt{2}} , 1)$
Let $f(x)=x+\frac 1x -\sqrt{2} ,\space x>0$ .Then show that $f(f(x))-x$ is monotonically decreasing on $(\frac 1{\sqrt{2}} , 1)$Seems a preety trivial question but have been trying for long without...
View Article$L^1$ convergence for product of arbitrary $L^1$ function with sequence of...
I'm studying for a qualifying exam, and I came across this question that I wasn't able to solve:Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of Lebesgue measurable functions defined on $[0, 1]$ such...
View ArticleLet $f:\Bbb R\to [0,\infty)$ be such that for any finite $E$, $\sum_{x\in...
Let $f:\mathbb R \to [0,\infty)$ be such that for any finite subset $E$ of $\mathbb R$,$$\sum_{x\in E} f(x)\leq1.$$Let $C_f=\{x \in\mathbb R \mid f(x)>0\}.$Then is $C_f$ is countable?How it is prove
View ArticlePointwise convergence of almost periodic function
This question is inspired by this post and definitions I use can be found there.Fix $f$ a real-valued almost periodic (but not periodic) function. For each $t\in\mathbb{R}$, define $f_t(x) = f(x-t)$...
View Articlecontinuous function question
Assume that function $f$ is continuous at $x=0$. Prove that the function $f(x)=a^x$for $a>0 $, is continuous at every real number.I know that $f$ is continuous at 0 if and only if 0 is in the domain...
View ArticleWhy N is their only model (up to isomorphism) in peano system?
Definiton(Natural number set):The set $N$ is nonempty. $0 \in N$, $S:N \to N$, if they meet the following condition:$S$ is injection , i.e. $S(y_1) = S(y_2) \Rightarrow y_1=y_2$$0 \notin S(N)$$N =...
View Articlesome uniform continuous functions
We need to find which are uniform continuous (UC) on a) $(0,1)$ and b) $(0,\infty)$. I have done, could you confirm me, if I am wrong any where?$\frac{1}{(1-x)}$ $\frac{1}{(2-x)}$ $\sin x$...
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