# Questions tagged [analysis]

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).

27,871 questions
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### Equivalence condition of Absolute Continuity

$f:I\to\mathbb R$ is continuous. Take finite number of subintervals $[x_1,y_1]\cup [x_2,y_2],...,\cup[x_n,y_n]=\mathcal K\subseteq I$, where $\sum_k\mu([x_k,y_k])=K$, i.e. the Lebesgue measure of all ...
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### Show that $\Lambda(t)\le-\frac C2\Lambda'(t)$ if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing

Let $\Lambda\in C^1([0,\infty))$ and $C>0$. Why does $$\Lambda(t)\le-\frac C2\Lambda'(t)\;\;\;\text{for all }t>0$$ hold if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing in $t$? Is this just ...
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### General rule for converting between sigma and pi notation

What is the general rule for converting from $\Pi$ to $\sum$ notation?
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### Proof of Osgood's Uniqueness Theorem

I am working on a proof of Osgood's Uniqueness Theorem. It is a somewhat guided exercise in which I am given intermediate steps to prove that should eventually result in a proof of the greater theorem....
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### Re-writing pi notation using exponentials

How might one show that the following is true by re-arranging the term on the left: $$\Pi_{r=0}^{n-1} e^\frac{2ri\pi}{n}=(-1)^{n-1}$$
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### Re-write exponent using sums [on hold]

What are all the possible ways one might re-write the following using the property $e^ae^b=e^{a+b}$: $$e^ \frac{2ri\pi}{n}$$
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### Sequence with convergent subnets but no convergent subsequences

We can regard a sequence as a special kind of net. But the definition of "subnet" is more flexible than that of "subsequence", so it's easy to find subnets of a sequence that aren't subsequences. In ...
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### Can this theorem be extended thus?

Consider the theorem: If $f$ is a continuous and differentiable real-function over an interval $I$ and $f'(x)\ge 0$ on $I,$ then $f$ does not decrease over $I.$ Somehow, I expect this to be true ...
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### What is wrong with the second proof?

Let $f$ be a real-valued function continuous on an interval $I$ and differentiable on every interior point of $I.$ Then if $f'(x)\ge 0$ everywhere inside $I,$ $f$ does not decrease over $I.$ There's ...
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### Differentiation and Conformity of a Vector-Valued Function

Let $r=n$ and g be a differentiable transformation. Then g is called conformal if there exists a real-valued function $\mu$ such that $\mu (t) > 0$ and $\mu (t) * D$g$($t$)$ is a rotation of $E^n$ ...
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### Walter Rudin's mathematical analysis: theorem 2.43. Why proof can't work under the perfect set is uncountable.

I found several discussions about this theorem, like this one. I understand the proof adopts contradiction by assuming the perfect set $P$ is countable. My question is if the assumption is $P$ is ...
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### A multiple choice question about $e^{-|x|}$

For $f:\mathbb{R}\rightarrow\mathbb{R}$, $f(x) = \exp (-|x|)$ Is the function a) bounded b) differentiable c) $f(\mathbb{R})$ is compact d) $f$ has a minimum According to the question only one ...