Find the domain of the functions of the type $f(x)^{g(x)}$.
Find the domain of $(\frac{2+x}{1-x})^{\frac{1}{x}}$I tried to find the domain of this function but could not find.Then i referred the symbolab.com domain calculator,and it showed me a technique to...
View ArticleA question about the relation between uniform continuous map and Cauchy...
Suppose $X,Y$ are metric spaces, $f:X\to{Y}$ is a map.If $X$ is compact metric space, we know $f$ is a uniform continuous map $\Leftrightarrow$ for all Cauchy sequence $x_n$, $f(x_n)$ is a Cauchy...
View ArticleProof of Sequential Criterion
Here is my attempt at a proof for the Sequential Criterion.Definition: Sequential Criterion for ContinuityA function $f: A \to \Bbb R $ is continuous at the point $c \in A$ if and only if $\forall$...
View ArticleInterval related to increasing/decreasing and concavity/convexity
Why do some people use closed intervals when describing the intervals where a function is increasing/decreasing or concave/convex?For example, given the function $f(x)= x^2-5x+6$, it says the interval...
View ArticleMotivation for Nets in topology
In Folland chapter 4 he provided an example for why sequential convergence doesn't play central role in general topological spaces as it does in metric spaces. he said consider $C^{\mathbb{R}}$, space...
View ArticleExample of apostol's sufficient condition for differentiability of...
I am talking about the theroem stated in : Mathematical Analysis, by Tom Apostol page $357$.$\textbf{Theorem.}$ Assume that one of the partial derivatives $D_{1}\mathbf{f},\cdots D_{n}\mathbf{f}$...
View ArticleConnectedness of intersection of two sets
Let $K$ be a compact subset of $\mathcal{R}^n$. Let $x_0\in \partial K$ (boundary of $K$). I want to show there exists a $\delta$ Such that $B_\delta(x_0)\cap K$ is connected. I don't even know if the...
View ArticleSolving $f'(x) = -g(x)/x$, $g'(x) = f(x)/x$
Find all differentiable functions $f$ and $g$ at $(0,∞)$ such that $f'(x) = -g(x)/x$ and $g'(x) = f(x)/x$ for all $x > 0$.I tried to determine the second derivative of $f$, then substituted $g(x)$...
View ArticleFor $z$ complex, do we have $\lim_{|z| < 1, \, z \to 1} (1 - z) \Big( \sum_{m...
Edit:It is well known that for real variable $x$, $\lim_{x \to 1^-} (1 - x) \Big( \sum_{m = 0}^\infty x^{m^2} \Big)^2 = \frac{\pi}{4}$.I would like to know whether this limit continues to hold if we...
View ArticleIf $\lim_{n \to \infty} a_n= \infty $ then there is a $n_0 \in \mathbb N $...
If $$\lim_{n \to \infty} a_n= \infty $$ then there is a $n_0 \in \mathbb N $ such that $ \forall n \ge n_0$$$a_{n+1} \ge a_n .$$Is that true? It seems true for me but I couldn't prove it or find an...
View ArticleConvexity conditions on $1^{st}$ and $2^{nd}$ derivatives at two points
Assume to have a smooth (or at least $C^2$) and convex real function $f$ such that:$f(x)>0$ for $x\in (a,b)$$f'(x)>0$ for $x\in (a,b)$$f''(x)>0$ for $x\in (a,b)$Now, assume that we are given...
View ArticleProving a function is constant, given $U(P, f) = L(P, f)$ for a certain...
The question goes as follows:Let $f : [a, b] → \mathbb{R}$ be a bounded function. Suppose thatthere is a partition $P$ of $[a, b]$ such that $L(P, f ) = U(P, f)$.Show that f is a constant function.I...
View ArticleProve $B:=\left\{\frac{y}{y-1}:y \in \mathbb{R} \setminus \{1\}\right\}$ is...
Given $$B =\left\{\frac{y}{y-1}:y \in \mathbb{R} \setminus \{1\}\right\}$$prove that $B$ is not bounded below nor above.I was thinking of using contradiction of the lemma below. Is that correct, or can...
View ArticleIntegration of softplus function
The softplus function $f(x)=\ln(1+e^x)$ is a good approximation for $\max\{x,0\}$.However, finding the primitive of $f(x)=\ln(1+e^x)$ seems to be quite difficult. Is there any tractable way to do that?...
View ArticleProve this set has zero g-density
$\omega=\{0,1,2,...\}$We define g-density for a set A like this:$$d(A)= \limsup_{n\to\infty} \dfrac{|A\cap \{1,2,...,n\}|}{g(n)}$$When $g: \omega \to [0,\infty) $ with $\lim\limits_{n \to \infty} g(n)=...
View ArticleConnectedness of intersections of sets
Let $K$ be a compact subset of $\mathcal{R}^n$ with nonempty interior. Let $x_0\in \partial K$ (boundary of $K$). I want to show there exists a $\delta$ such that for any $\delta'<\delta$,...
View ArticleConvexity of a set that contains a nonempty convex compact set [duplicate]
Let $K\subset\mathcal{R}^n$ be nonempty, convex and compact. Define the distance from a point to $K$ as $d(x,K)=\inf_{y\in K}d(x,y)$. Let $W=\{x\in \mathcal{R}^n\colon d(x,K)<r\}$. By drawing little...
View ArticleUsing Hardy-Littlewood inequality to prove a theorem about interleaving...
Fix a positive integer $k$. For each $i\in[k]$ let $g_i=(g_it)_{t=1}^\infty$ be a sequence of numbers. Given an infinite sequence $(a_t)_{t=1}^\infty$ taking values in $[k]$ define the interleaving...
View ArticleA problem about Fubini's Theorem and norm
Let $f \in L_{1}(\mathbb{R})$ and for each $h > 0$, we defined: $$f_{h}(x) = \frac{1}{2h} \int_{(x-h,x+h)} f(t) \lambda(dt),$$ where $\lambda$ is the Lebesgue measure on $\mathbb{R}$.I want to prove...
View ArticleExamples of smooth functions with 'pathological' properties
In real analysis, it seems that for every nice condition you might expect to hold, there exist numerous counterexamples, often with creative constructions. For example, the Weirstrass function shows...
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