# Tagged Questions

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### Banach Fixed Point Theorem Incomplete Conditions

let $X = [0,\infty)$ equipped with the standard metric $d(x,y) = |x-y|$. Let $f: X \rightarrow X$ be defined by $f(x)=x+e^{-x}$ Explain why this function doesn't contradict Banach's Fixed Point ...
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### Why doesn't exist a Cousin's lemma for left-tagged partitions?

I am thinking of the possible validity of a statement like this: given any positive mapping $\delta$ on $[a,b]$, there exists a partition $\, a=a_0<a_1<\cdots<a_n=b \,$ of $\,[a,b] \,$ so ...
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### Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
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### A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
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### A die is thrown 10 times. What is the probability that $6$ isn't registered and that at least one “1” is registered.

$A$, first occurrence - that $"6"$ isn't registered $B$, second occurrence - that at least one $"1"$ is registered. What I know: How to find $P(A)$ and $P(B)$ (over their complements) What I'm ...
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### orthogonal polynomials: explicit representation

Consider a sequence of orthogonal polynomials $P_0(x) = 1$, $P_1(x) = x$, and recursively $P_{n}(x) = (a_n x + b_n) P_{n-1}(x) + c_n P_{n-2}(x)$ for some sequences of real constants $a_n$, $b_n$, ...
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### Collection of useful inequalities

I was wondering if anyone knows of a good list of useful inequalities? If not, I'd like to compile one here for reference. For example, to prove $$\sum_{n \in N} \frac{\sqrt{a_n}}{n}$$ is convergent ...
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### Does $f ^{-1}( x )$ continuous if $f(x)$ is a continuous function and $f'(x)>0$ [duplicate]

If $f(x): [a,b] \to \mathbb{R}$ is a continuous function and $f'(x)>0 \ \forall x \in (a,b)$. Dose $f^{-1}(x)$ continuos ?
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### Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
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### Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
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### Continuous functions as regulated functions: a property.

In Differential and Integral by Paul Lorenzen (1971) pag. 148, I read ... every continuous function is trivially approximable by step functions that have no jump at a given arbitrary point .... All ...
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### The range of the derivative of a differentiable function

I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be (1) a closed interval or (2) an open interval or (3) a half-open interval or (4) an unbounded interval ...
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### The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
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### Cauchy in metric space

Let $C[-1,1]$ be the space of continuous functions with metric $$\rho(f,g)=\max\{|f(x)-g(x)|: x\in [-1,1]\}\;.$$ Then the sequence of functions $(f_n) :[-1,1] \to \Bbb R$ defined as ...
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### contraction metric space

"Let $0 < a < 1$ and $f(x) = (x^2 + a)/2$. Show that $f : [0, a] \to [0, a]$ and that f is a contraction. Find the fixed point of $f$." For this problem, since there isn't any metric $d(x,y)$ ...
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### determine whether a metric space is complete or not

How to decide if the metric spaces $((0,1)$, $d(x,y)=|x^2-y^2|)$ and $((-\frac{\pi}{2},\frac{\pi}{2})$, $d(x,y)=|\tan x-\tan y|)$ are complete or not. For the first metric, I let any cauchy sequence ...
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### A dense subset in $\mathbb{R}^n$

I have some difficulties in the following question. I would like to thank for all comments and kind help. Let $S$ be a dense subset in $\mathbb{R}^n$. In this case, whether we can find a straight ...
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### How to show that continuous functions between metric spaces agree on a closed set

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that: If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.
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### Property of strictly convex polynomial

I have some difficulties in the following problem. Thank you for all comments and helping. Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial. Suppose that $f$ is strictly ...
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### Are the non-standard models of classical analysis the standard models of non-standard analysis?

I've never quite understood how the non-standard models of classical analysis related to the standard models of non-standard analysis. I know that non-standard analysis got its start with Robinson ...
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### equivalent metric

Let $(X; d)$ and $(Y; d')$ be metric spaces, and let $f : X \to Y$ be continuous. Define $df (x; y) = d(x; y) + d'(f(x); f(y))$ for $x, y \in X$. Show that $df$ is a metric on $X$ that is equivalent ...
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### Question on relations [duplicate]

-1 down vote favorite Define the relation on Q by [m,n]<[j,k] if and only if jn−mk belongs to N, j and m belong to Z, n and k belong to N. (a) Show that < is well defined, that is if ...
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### Question about relation on rational numbers

Define the relation on $\mathbb{Q}$ by $$[m,n]<[j,k]$$ if and only if $jn-mk$ belongs to $\mathbb{N}$, $j$ and $m$ belong to $\mathbb{Z}$, $n$ and $k$ belong to $\mathbb{N}$. (a) Show that ...
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### Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
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### Directional differentiable property

I am stuck in constructing a function that is locally Lipschitz continuous at $x_0$ but it does not have directional differentiation at $x_0$in any direction. Thank you for all help and comments.
I am looking for a reference for the following fact. Any hints would be appreciated. Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...