# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Alternatives to Chapters 8-10 of Rudin's PMA

S.E advisers, I have been hearing that the chapters on multi-varaible analysis in Rudin's PMA are almost nothing like previous insightful chapters in the single-variable analysis, and I verified ...
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### Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

Is it possible to estimate the following Lebesgue integral ($\|\cdot\|$ is the 2-norm) $$\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x, \, x\in\Bbb R^d$$ in terms of $\delta$ when $\delta\to 0$? ...
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### Compact set intersects finitely many elements of an open set covering of a bounded open set prove [closed]

I saw @MISC {1697170, TITLE = {Compact set intersects finitely many elements of an open set covering of a bounded open set}, AUTHOR = {Hua (http://math.stackexchange.com/users/269509/hua)}, ...
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### Intermediate value property with no continuity

Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) < v < f(b) or f(b) < v < f(a); that is, if v is between f(a) and f(b), there is ...
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### Why is this function smooth on the coordinate axis

Consider the function $$f(x,y):=\sqrt{x^2+xy+y^3}, \quad x,y \geq 0.$$ It is claimed that this function is smooth except at the origin. I am wondering why this function is not smooth at (0,0) in the ...
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### Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$

I want to prove that all "closed unit ball" of $$c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$$ do not have any extreme point. Would you please help me? (Extreme ...
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### Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$\mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?.$$ No ideas how to start this one. I see that the limit of ...
Let $f:[0,1] \rightarrow R$ defined by $f(x) = 2$ if $x = \frac{1}{n}$ for some $n∈ℕ$, $0$ otherwise. Determine if $f$ is Riemann integrable. My attempt: Let $ε > 0$. Construct a partition P as ...