# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Proving the limit at $\infty$ of the derivative $f'$ is $0$ if it and the limit of the function $f$ exist. [duplicate]

Suppose that $f$ is differentiable for all $x$, and that $\lim_{x\to \infty} f(x)$ exists. Prove that if $\lim_{x\to \infty} f′(x)$ exists, then $\lim_{x\to \infty} f′(x) = 0$, and also, give an ...
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### Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
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### Order of study? Rudin, Spivak, Munkres?

I'm currently taking an analysis course at a top 10 four year university in which we use Baby Rudin as our primary text. I was curious to know the order in which I should continue my studies. That ...
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### Is $[0,1]^2 \setminus \{(a,b)\}$ connected?

I am pretty sure that this set is in fact connected but I am struggling to see how to prove it, it is simple to see that $[0,1] \setminus \{x\}$ is disconnected but I can't see how to relate ...
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### The dimension of the operator if the domain has dimension 2

Suppose $A$ is a linear operator s.t. $A\colon X\rightarrow Y$. If $\dim(X)=2$, what is $\dim(A(X))$?
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### “Greatest lower bound function”

If $f$ is a function continuous at $c, h$ is positive and $m$ is a function defined as $m(h)=\inf \{ f(x): x \in [c,c+h] \}$ , how can I prove that the limit of $m$ as $h$ approaches $0$ ...
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### Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
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### Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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How to prove that $$f :E\rightarrow F ~\text{is open} \Longleftrightarrow f^{-1}(\overline{A})\subset \overline{f^{-1}(A)}, \forall A\subset F$$ where $(E,\tau), (F,\theta)$ are topological spaces. ...
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### Prove the following about absolutely convergent complex series

Prove that for every sequence $(a_n)_n$ of complex numbers, if the series $\sum_{n\ge 0} a_n$ is absolutely convergent, then $|\sum_{n\ge 0} a_n| \le \sum_{n \ge 0} |a_n|$. I've been given the ...
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### How does Wolfram Alpha come up with the substitution $x = 2\sin u$? Integration/Analysis

I have to integrate $$\int_0^2 \sqrt{4-x^2} \, dx$$ I looked at the Wolfram Alpha step by step solution, and first thing it does is it substitutes $x = 2\sin(u)\text{ and } \,dx = 2\cos(u)\,du$ ...
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### How to compute $\lim_{n\to \infty} n\sin(2\pi n! e)$ [duplicate]

I want to calculate $$\lim_{n\to \infty} n\sin(2\pi n! e)$$ I have used the Stirling approximation and I think the answer is zero . But I think the limit maybe not exists. Can some one help? ...
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### Continuous function positive at a point is positive in a neighborhood of that point

Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) > 0$ then there exists a $d > 0$ such that $\forall x$, $f(x) > 0$ with $|x-c| < d$. I can understand ...
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### Prove exponent(m)=e^{m}

please show me how to do the third one, I just understand the 1st and 2nd, but i have no idea how to do the 3rd. thank you.
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### bounded interval is bounded and connected

Can you please tell me if my proof is correct? Definition: Let $X$ be a subset of $\mathbb R$. We say that $X$ is connected iff the following property is true: whenever $x, y$ are elements in ...
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### local maxima of weierstrass function

Does the weierstrass function have uncountably many local maxima on (0,1)? I don't really know how to approach this problem at all, so any help is appreciated
### If $f$ and $g$ are continuous, then max(f, g) is continuous and differentiable
If $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then $\max(f, g)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. I'm asked to either prove or disprove this ...
Let $0< a_1 < b_1$ and define $a_{n+1} = \sqrt{a_nb_n}$, and $b_{n+1}$ = ${a_n + b_n}\over2$. Use induction to show that $a_n<a_{n+1}<b_{n+1}<b_n$. Also, prove $a_n$ and $b_n$ ...