Mathematics Stack Exchange
Mathematics Stack Exchange

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Shift space without any periodic points. An irrational number with a bounded number of consecutive 0's?

https://en.wikipedia.org/wiki/Shift_space for definition of a shift space. My first attempt was to take an infinite binary string, in which every word occurs in it. Then close it under the shift operator, however this space is not closed, and…
Mars
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Lebesgue Integration Proof

I have been trying to teach myself Lebesgue Integration and came across the following question in a textbook. Let $g \subset [0,1]$ be bounded and measurable. Suppose that it satisifes the property $\int_{[0,1]} g \chi_{[0,c)}\, d\mu = 0$ for all $c…
John
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Proving that a subset $\mathscr{H}$ is dense in $\mathbb{R}$

Consider the subset $\mathscr{H} =\displaystyle\left\{\frac{a}{10^{b}} \ |\: a,b\in\mathbb{Z}\: \right\}$. Then is $\mathscr{H}$ dense in $\mathbb{R}$. To prove that $\mathscr{H}$ is dense I have to show that given $x,y \in\mathbb{R}$ there is a…
dumbo
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Some questions regarding open sets and its complement.

Let $E^o$ be the set of all interior points of the set $E$. I was able to prove that $E^o$ is always open and $E$ is open iff $E = E^o$. Now I am asked to prove that $(E^o)^c = \overline {E^c}$. Intuitively it is very clear, but I am not sure if my…
hyg17
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Let ${(a_n)}$ be a bounded sequence that does not converge.

Let ${(a_n)}$ be a bounded sequence that does not converge. Show that ${(a_n)}$ has two subsequences that converge to different limits. I think I am supposed to prove $(a_n)$ does not converge and for some reason all I can think of doing is to show…
user102156
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Proof on showing a uniformly continuous function has limit at every cluster point of the domain

The question is as follows: Given: (a) f is uniformly continuous on a subset D of $\mathbb R^n$ and (b) $x_0$ is a cluster point of D Show: The limit of f(x), as x approaches $x_0$, exists in D Here are my attempts: Attempt 1: Direct proof 1/…
Cecile
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Confusion about proof in Spivak?

I am currently reviewing some of the proofs in Spivak's Calculus, and I have come across this statement as a part of the proof for Extreme value theorem. if $f(x)$ is continuous on $[a, b]$ then for every $\epsilon > 0$ there is $x$ in $[a, b]$…
Mark
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Consequence of inequality

My question: Let $\Omega$ in $\mathbb{R}^n$ bounded. For all $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that $$ \label{lemma_gagliardo_nirenberg_2} \|\varphi\|_{L^2(\Omega)}^2 \le \varepsilon \|\nabla…
mathmatix
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periodicity of function

If $f(x+1) + f(x-1) = \sqrt3f(x)$, then what is the period of $f(x)$?
mun
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given two line segments AB and CD, it would always be possible to find a third line segment whose length divides evenly into the first two.

What does it mean that given two line segments AB and CD, it would always be possible to find a third line segment whose length divides evenly into the first two ?? Does it means that the third line segment is AB + CD ?? It comes from the book…
ju so
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Connected manifold

I need to prove a rather simple fact: let $X$ be a connected smooth (or just topological) manifold. Now we need to prove that $X$ is linearly connected. So, it seems to be obvious, because $X$ is locally like a unit open ball in $\mathbb{R}^n$, but…
user74574
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Analysis involving closed intervals and rationality

Can someone help me prove this problem? Prove that every closed interval $[a,b]$ is a subset of $\mathbb{R}$ contains at least one rational number.
miguel barnes
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Uniform continuity of composition?

If we consider $f\circ u$ and know that f is bounded and uniformly continuous u is bounded and continuous does this imply that $f\circ u$ is bounded and uniformly continuous? It is clear that it is bounded. But it is not clear to me whether it is…
Salamo
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Finding $\lim_{x \to +\infty} \left(1+\frac{\cos x}{2\sqrt{x}}\right)$

Let $f(x)=x+\sin(\sqrt{x})$. I want to find $\lim_{x \to +\infty} f'(x)$. Attempt 1 We have $$f'(x)=1+\frac{\cos x}{2\sqrt{x}} \leq 1 + \left|\frac{\cos x}{2\sqrt{x}}\right| \leq 1 + \frac{1}{2\sqrt{x}}.$$ As $x \rightarrow \infty$, $\sqrt{x}…
user4167
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proving a set is not path-connected

Let $$A=\{(x,\sin(1/x)) \mid x > 0 \} \cup \{(0,y) \mid y\in[-1,1]\} \subset\mathbb{R}^2$$ how to prove that this set is not path-connected(or just not connected)? Thank you
user74311
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