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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Can the act of Taylor expansion be written as an exponential?

For example: $f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2!}f''(x_0)(x-x_0)^2+\dots=\exp\left((x-x_0)\frac{\mathrm{d}}{\mathrm{d}x}\right)f(x)\Big|_{x_0}$ I don't know how to write the $\Big|_{x_0}$ to the front … Is this thing an operator? How to write it…
LePtC
  • 143
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4 answers

Is it true that every bounded sequence with the following property converges?

Is it true that every bounded sequence $\{a_n\}$ of real numbers such that $|{a_n - a_{n-1}}|<1/n$ for all $\ge2$ is convergent?
Nomis
  • 21
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Prove that the set of polynomials in C([a, b], R) is not open

So I'm trying to prove that the set of polynomials in C([a,b],R) is not open. I understand the definition of an open set, but I'm wondering how to find a point that is not contained in the interior if it's the set of polynomials. Also, Can a…
user146296
  • 139
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In which metric spaces other than the discrete spaces are the closures of open balls different from closed balls?

Let $(X,d)$ be a metric space such that $d$ is not the discrete metric. Let $x_0 \in X$, let $r>0$, and let $$B(x_0;r) \colon= \{ x \in X \colon d(x,x_0) < r \}$$ be the open ball with center $x_0$ and radius $r$, and let $$\tilde{B}(x_0;r) \colon=…
Saaqib Mahmood
  • 26,762
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2 answers

For a natural number $n$ and numbers $a$ and $b$ such that $a \geq b \geq 0$, prove that $a^n-b^n \geq nb^{n-1}(a-b)$

I tried to do an induction proof and I've played around with it for about an hour and haven't really gotten anywhere. For my base case, I let $n=1$ and got $(a-b) \geq (a-b)$, however when I tried to show $a^{k+1}-b^{k+1} \geq…
user3034400
  • 145
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How to show that a strictly decreasing, continuous function which decays slower than 1/x is not integrable?

I want to show that a function that decays slower than $1/x$ is not integrable and I tried it the following way: Assume the positive, strictly decreasing and continuous function $g(x)$ decays slower than $1/x$, i.e. $$xg(x)\to\infty\,\,…
Tim
  • 109
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Cauchy sequences.

Well my question is how to prove this: Let $a_{0}$, $a_{1}$ be distinct real numbers.Define: $$a_{n}=\frac{a_{n-1}+a_{n-2}}{2}$$ for each positive integer $n\ge2$.Show that $\{a_{n}\}$ is a Cauchy sequence. (Hint: You may want to use induction to…
user162343
  • 3,245
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Construct continous function that passes through a countable set of points

Suppose I have two sequences, $t=(t_1,t_2,... )$ in $[0,1]$ and $y=(y_1,y_2,...)$ in $\mathbb{R}$. Is it possible to construct a continuous function $f:[0,1]\longrightarrow \mathbb{R}$ such that $f(t_i)=y_i$ $\forall i \in \mathbb{N}$? If it is…
Vokram8
  • 341
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How to show $\textrm{supp}(f*g)\subseteq \textrm{supp}(f)+\textrm{supp}(g)$?

Let $f, g\in C_0(\mathbb R^n)$ where $C_0(\mathbb R^n)$ is the set of all continuous functions on $\mathbb R^n$ with compact support. In this case $$(f*g)(x)=\int_{\mathbb R^n} f(x-y)g(y)\ dy,$$ is well defined. How can I show…
PtF
  • 9,655
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2 answers

Is it true that for locally $L^p$ integrable function, $\int_{[a,b]} |f(x+y)-f(x)|^p dx \to 0$ as $y \to 0$?

If $f \in L^p(\mathbb{R})$, where $p\in [1, \infty)$, then $$\int_{-\infty}^\infty |f(x+y)-f(x)|^p dx \to 0 \ \textrm {as} \ \ y \to 0.$$ Assume now that a function $f: \mathbb{R} \to \mathbb{R}$ is locally in $L^p$, that is for each compact…
A.B
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7 answers

Prove that if $n$ is an integer, then $n^2 + n^3$ is an even number

I am trying to work through some of the problems in Stephen Lay's Introduction to Analysis with Proof before my Real Analysis class in the fall term starts, and I was just wondering if I could get some feedback as to whether or not I have completed…
MathMajor
  • 6,478
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1 answer

Importance of specifying indexing sets

I was going through a rudimentary course in mathematical analysis covering Metric spaces and the book opens up with the idea of open sets. While mentioning the property of open sets it cites that if I is any indexing set then union of open sets Gi…
user12817
  • 21
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Can non-constant functions have the IVP and have local extremum everywhere?

Let $f:\mathbb R \to \mathbb R$ has Intermediate value property. If f has local extremum at every point of $\mathbb R$, can we say f is constant? We know $$f(x)=\begin{cases}1 & x \in \Bbb{Q} \\ 0 & x \not \in \Bbb{Q}\end{cases}$$ has I.V.P and…
Fin8ish
  • 3,361
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0 answers

Absolutely continuous function admits weak derivative

How to prove that an Absolutely continuous function admits weak derivative? Absolutely continuous function: Let $(X, d)$ be a metric space and let I be an interval in the real line R. A function $f: I → X$ is absolutely continuous on I if for every…
Rahman
  • 21
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1 answer

Constructing a set that contains at most one point on vertical and horizontal

I'm not sure how to answer this question: Construct a set $A$, which is a subset of $[0, 1] \times [0, 1]$, such that $A$ contains at most one point on the horizontal and vertical lines, but boundary $(A) = [0, 1]\times [0, 1]$. Is the answer just…
william
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