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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+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
23
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1answer
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Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers ...
2
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1answer
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A sequence of continuous functions which is pointwise convergent to zero and not uniformly convergent on any interval.

The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly ...
3
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0answers
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How do find the numerical average of $x^x$ from $(-4,-2)$ without x-values that give a complex output?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all real inputs that give a complex output and need average to be a real number. To ...
1
vote
2answers
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+150

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most ...