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
3
votes
1 answer

Help understanding Dedekind cuts

I have an exam this tuesday and our prof gave us these problems to practice. Me and my friend were trying to do it. but I really never understood the concept or if its right. These are the questions: 1) Let $A_1$ be the set of rational numbers $x$…
MathGeek
  • 886
3
votes
1 answer

How to prove that $C = \bigcup_{a \in A \times \{0\}} \{(1-t)a+tv| 0\leq t \leq 1\}$ is Jordan-measurable if $A$ is Jordan-measurable?

Let $A \subseteq \mathbb{R}^{n-1}$ be Jordan-measurable and $$C = \bigcup_{a \in A \times \{0\}} \{(1-t)a+tv| \; 0\leq t \leq 1\}$$ be an object in $\mathbb{R}^n$ with the base area $A \times \{0\}$ and the apex $v \in \mathbb{R}^n$. I have to show…
Huy
  • 6,674
3
votes
1 answer

Exercise 4.3.12 in Understand analysis

Let $F \subset R$ be a nonempty closed set and define $g(x) = \inf \{ | x - a | : a \in F \}$. Show that g is continuous on $R$. I follow a solution, and this is its approach to prove the statement. First, it proves $\forall x \in R$, there exists…
Jesse
  • 183
3
votes
0 answers

Intuition behind upper and lower hemicontinuity

I've been studying quite a bit about upper and lower hemicontinuity with reference to correspondences. However im having a hard time seeing the advantage of this definition and what it adds to the overall understanding of continuity in…
EconJohn
  • 165
3
votes
1 answer

Analysis Convergence/Divergence

Prove there exists a function $f$ such that $$\int_1^{\infty}f(x)\,dx\text{ converges, but }\int_1^{\infty}|f(x)|\,dx\text{ diverges.}$$ Similarly, prove that there exists a function $g$ such that $$\int_0^1 g(x)\,dx\text{ converges, but…
mary
  • 2,374
3
votes
2 answers

$f^n(x)$ where $n \in \mathbb{R}^+$

Given a function $f(x)$ whose image is a subset of its domain, we can define $$ f^n(x) = \underbrace{f(f(f(\dots f(x) \dots )))}_{n \text{ times}} $$ This makes sense when $n$ is a nonnegative integer. Can we extend this definition to continuous…
Alex Van de Kleut
  • 161
3
votes
1 answer

Prove there are distinct $x_1,\,x_2,\cdots,\,x_n$ such that $ \sum_{i=1}^n\frac{p_i}{f'(x_i)}=\sum_{i=1}^n p_i. $

Suppose $f(x)$ is differentiable on $[0,\,1]$, $f(0)=0$, $f(1)=1$ and $p_1,\,p_2,\cdots,\,p_n$ are $n$ positive real numbers. Prove there are distinct $x_1,\,x_2,\cdots,\,x_n$ such that $$ \sum_{i=1}^n\frac{p_i}{f'(x_i)}=\sum_{i=1}^n p_i. $$ I…
Knt
  • 1,649
3
votes
0 answers

Rigorous hypothesis for Reynolds' transport theorem

I'm looking for rigorous hypothesis for the application of Reynolds' transport theorem : $$ \frac{\mathrm{d}}{\mathrm{d}t}\left[ \int_{\Omega(t)} \phi({\bf x},t) \mathrm{d}{\bf x} \right] = \int_{\Omega(t)} …
Vincent Nivoliers
  • 665
3
votes
3 answers

Why this function is not uniformly continuous?

Let a function $f: \mathbb R \rightarrow \mathbb R$ be such that $f(n)=n^2$ for $n \in \mathbb N$. Why $f$ is not uniformly continuous? Thanks
Richard
  • 4,432
3
votes
1 answer

Prove that $1+rx = (1+x)^r$ holds only if $x=0$.

I believe my work is correct here, just looking for closure. Is it true that since r>1, the last line could be (1+x)^1? Suppose $1+rx\le(1+x)^r$ for any real number $r>1$ and $x>-1$. Prove that the equality holds only if $x=0$. My work: For…
user65384
  • 614
3
votes
1 answer

Finding x and y for continuous onto function

$f:[0,1]\rightarrow [0,1]$ continuous onto function such that $f(0)=0=f(1)$. Then show that there exists distinct values $x,y$ belongs to $[0,1]$ such that $f(x)=\frac{1}{2}=f(y)$. could you please tell some hint?
prasad
  • 517
3
votes
4 answers

Find the asymptote of the function $f(x) = \sqrt{\frac{x^3}{x - 3}} - x$

We have a function $f(x) = \sqrt{\frac{x^3}{x - 3}} - x$ and when $x$ goes towards $-\infty$, we have an asymptote $y = -2x - 3/2$. How we get this asymptote?
Niki Sedlarević
  • 531
3
votes
2 answers

$h(x)=\min\{f(x),\,g(x)\}$, it's possible that $\int_0^{\infty}h(x)\,dx<+\infty$? $\int_0^\infty f(x)\,dx,\;\int_0^\infty g(x)\,dx$ are divergent.

Suppose $f(x),\,g(x)\in C([0,\,\infty))$ which are nonnegative and decreasing, such that $\displaystyle \int_0^\infty f(x)\,dx=\int_0^\infty g(x)\,dx=+\infty$. Let $h(x)=\min\{f(x),\,g(x)\}$, it's possible that…
Knt
  • 1,649
3
votes
1 answer

Prove that $I(t)=\int_{0}^{+\infty}x^tf(x)\,dx $ is defined on $(-1,\,1)$ and has continuous derivative.

Suppose the improper integral $\displaystyle\int_{0}^{+\infty}xf(x)\,dx$ and $\displaystyle\int_{0}^{+\infty}\frac{f(x)}{x}\,dx$ are both convergent, prove that $$ I(t)=\int_{0}^{+\infty}x^tf(x)\,dx $$ is defined on $(-1,\,1)$ and has…
Knt
  • 1,649
3
votes
0 answers

Copies of finite sets in sets of positive measure

We say a set $A \subseteq \mathbb{R}^n$ contains the pattern of a finite set $B \subseteq \mathbb{R}^n$ if there exists a shift $t \in \mathbb{R}^n$ and scale $s > 0$ such that $t+sB \subseteq A$. I've read that if $A$ has positive measure, then for…
Mark
  • 31
  • 1
1 2 3
…
99 100