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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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Showing that a certain function is a local diffeomorphism

I have to show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2: (x,y) \mapsto (e^x(x \cos y - y \sin y),e^x(x \sin y + y \cos y)$ is a local diffeomorphism in ever point not $(-1,0)$. I have no idea how to invert $f$ on some restricted domain. My…
Nga
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If $f,g$ each have property p then $f \circ g$ also has property $p$

What type of characteristics should properties have for the following to hold true? If $f,g$ each have property $p$ then $f \circ g$ also has property $p$ Examples: If $f,g$ continuous then $f \circ g$ is continuous If $f,g$ entire then $f \circ…
jimjim
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Compactness of unit ball in weak-operator topology.

I'm reading Richard Kadison's book about operator algebras, and in the demonstration that the unit ball is compact in weak-operator topology, the author defines a function from the set of bounded operators on a Hilbert space $H$, to a product of…
Felipe Pérez
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Prove that there is no $1-1$ continuously differentiable map $f : \mathbb{R}^2 \to \mathbb{R}.$

Possible Duplicate: existence of a map between $\mathbb R^2$ and $\mathbb R$ Prove that there is no $1-1$ continuously differentiable map $f : \mathbb{R}^2 \to \mathbb{R}$. I know it's something to do with the implicit function theorem but I…
Denis
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Lipschitz maps on $[0,1]$

Here is just a little curiosity. Assume the $f : [0,1] \to [0,1]$ is a Lipschitz function that maps 0 to 0 and 1 to 1. If we impose that the Lipschitz constant of $f$ is $\le 1$, can $f$ be anything other than the identity map?
passerby51
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Help with proof by contradiction

I'm relatively new with proofs and am trying to self-teach. I'm currently going through questions that unfortunately have no solutions... I've been doing well until I struck this one: If l, m, and n are consecutive integers, then 12 does not divide…
4yourneed
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Proof of multiplicative commutativity for all real numbers

I have seen proofs for commutativity for all integers, and these can be extended to rationals easily because a rational number is just the ratio of two integers. However, I have yet to see a proof that multiplication of real numbers is commutative.…
u8y7541
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Rearrangement of series in a (not necessarily Banach) normed vector space.

If $X$ is a Banach space, then the following theorem holds: Let $\sum x_n$ be a series in $X$ which converges absolutely. Then every rearrangement $\sum x_{\sigma(n)}$ converges, and they all converge to the same value. Proof: Let $(s_n')$ be the…
Gabriel
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Proving the existence of a point $a \in \mathbb{R}_+$ s.t. $\cos(a) < 0$

I am currently working on a challenge problem where I need to show that there is a point $x \in \mathbb{R_+}$ such that $\cos(x) = 0$ using only a few properties of the cosine function. In particular, the only properties of the cosine function that…
Elements
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Proving limit of sin(1/x)cos(1/x) doesn't exist as x goes to 0

Just a quick question, this may or may not be a duplicate by the way. I've seen the proof of the trig functions not existing separately but I couldn't seem to find them multiplied together like in this problem. My question is could I just separate…
Sky
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Distance between sets with one closed set and one compact set

I would like to prove the following statement. Let A and B be nonempty disjoint subsets of $\mathbb{R}$ where A is closed and B is compact. Then there exists $a \in A$ and $b \in B$ such that $inf${$|a-b|$:$a \in A, b \in B$}=$|a-b|$ A and B…
Tim Lee
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Proving Cauchy when given a sequence

Let $\left\{x_n\right\}$ be a sequence and $0 < a < 1$. Suppose that for all $n \ge 3$ we have $$ \left\lvert x_n - x_{n-1}\right\rvert \le a\left\lvert x_{n-1} - x_{n-2} \right\rvert. $$ Prove that $\left\{x_n\right\}$ is Cauchy. I don't even…
Regios
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Piecewise continuous vs. continuous

I'm a little confused by the concept of "piecewise-continuous". My understanding is that if a function $f$ is piecewise-continuous on some domain $[a, b]$, it is not necessarily continuous on $[a, b]$, but it can be cut into pieces which are…
Nażysław Zbyłutowicz
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lebesgue density theorem for smooth manifold (revised)

My question is about lebesgue density theorem: Let $\mathcal{H}^s$ be $s-$dimensional Hausdorff measure. If $A\subset \mathbb{R}^{n}$ with $0<\mathcal{H}^s(A)<\infty,$ then for $\mathcal{H}^{s}$ almost all $x\in A,$ $$\limsup_{r \rightarrow…
user44174
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How to prove that $\gamma'(n+1)\gamma(n+1)$ is incremental

Not so long before I asked this question, I encountered a question that asked me to prove a proposition, but I'm not sure how to go about it. Prove that $\gamma'(n+1)\gamma(n+1)$ is incremental
Green Liu
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