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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A question about continuous functions

Let $f:B \longrightarrow \mathbb{R}^n$ a continuous and injective function from closed ball in $\mathbb{R^n}$ to $\mathbb{R}^{n}$. I'd like to know $f$ has to maps the boundary $\partial B$ in the boundary $\partial f(B)$? Thank you.
user29999
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If f is differentiable with a continuous derivative function, then the set of critical points of f is closed.

If f is differentiable with a continuous derivative function, then the set of critical points of f is closed. Is this a true statement? I'm kinda lost.
user149131
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What locally integrable function $f$ satisfies $\int_a^ b f(x) \phi'(x)dx=0 $ for each $\phi \in C_0^\infty(a,b)$

Let $f:(a,b) \rightarrow \mathbb{R}$ be locally integrable and such that $$\int_a^ b f(x) \phi'(x)dx=0 \textrm{ for each } \phi \in C_0^\infty(a,b).$$ How to show, without help of distribution theory, that $f=const$ a.e.? I noticed that every…
Richard
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Prove that an alternating series converges to a value between the first term and the sum of the first two terms

This is used in an elementary proof that $e$ is irrational. I can prove this, but what I am doing is not particular nice looking. In the proof, the author says this is obvious but I can't seem to write it out so simply. \begin{equation} S =…
jlc1112
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About example of continuous function on $\mathbb{R}$ which cannot be uniformly approximated by polynomials?

Possible Duplicate: Weierstrass approximation does not hold on the entire Real Line If a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is continuous then $f$ can be uniformly approximated by smooth functions (see here). By the Weierstrass…
Richard
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Example for point wise convergence I don't understand

I am little bit confused on this example of point wise convergence. For $x \in [0,1]$ and $ n >= 2$ define $f_n(x) = \left\{ \begin{array}{lr} n^2x^2 &,\: if \: 0 <= x <=1/n \:,\\ -n^2(x - 2/n)&,\: if \: 1/n <= x <= 2/n\:,\\ …
user111750
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Definition of Borel sets

MathWorld says: Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class of Borel sets in Euclidean is the smallest collection of sets…
badatmath
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The strict convexity of the$ L^2$ norm

I am sucked by a very simple question. I want to find an element which minimizes the$\|A\|$(in $L^2$-norm ),where A is a random variable. Then the textbook says the uniqueness will follow from strict convexity of the $L^2$ norm. What does this…
Alex
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Rearrange equation with sums

I'm currently doing some physics and while trying to solve a problem I came across an equation that I can't rearrange for the variable (the question is pure mathematics so I'm asking it here). The equation is: $$H = \frac{\sum\limits_n^N E_n \exp (…
user44789
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How can an interval be an open ball?

I have been asked this question, but I do not understand it. "For what values of $\alpha$ and $\beta$ are the sets $(\alpha, \beta)$, $[\alpha, \beta)$, $(\alpha, \beta]$ and $[\alpha, \beta]$ open balls in the metric space $[a, b]$?" We have our…
frightenedeyes
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Find $\inf$ and $\sup$

Find $\inf$ and $\sup$ of $A=\left\{ \dfrac{2013}{1+\epsilon+\epsilon^{-1}}: \epsilon \in (0,1)\right\}$ . Check if $A$ has the biggest element and the smallest element.
gunia6
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Is open proper map surjective?

I want to know the relationship between "proper" and "surjective", is open proper map surjective? Or it needs more condition to imply surjection? For example, the map is a homogeneous complex map. Thanks in advance.
Yui
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Solutions to Biler and Witkowski

I am currently working through Biler and Witkowski's book, Problems in Mathematical Analysis - which I have found to be one of the greatest problem book in analysis that I've ever opened. I was wondering if anyone could point my in the right…
wxm5539
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how to show equivalence of arbitrary chosen element of a set.

Given a set $S\in \mathbb{R}$, let us write $−S$ for the set $\{−x \mid x\in S\}$. Prove that if $S$ is bounded below then $−S$ is bounded above. This is not a hard problem, but I am puzzled by the follow: I assumed $x$ is an arbitrary element in…
Kun
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Proof the formula

I have to prove the following formula: $$\sum_{k=0}^n \frac{(-1)^k}{k+1} \binom{n}{k} = \frac{1}{n+1}$$ I do have absolutely no clue ye about how to even start. I'm thinking about using binomial theorem, but how? Edit: Thanks to all the answers, but…
KevinSkyba
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