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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Pointwise convergence counter example.

Can anyone think of a counter example that for $f_n:[a,b] \to \mathbb{R}$ regulated and $f_n \to f$ pointwise but $f$ is not a regulated function? Thanks!
user26069
4
votes
1 answer

Analysis Constructing a Sequence

I'm looking for a sequence of functions that is continuous and absolutely integrable, but pointwise divergent for every $z$ $\in [0,1]$. In other words, $ \int_0^1 |f_n(z)| dz \rightarrow 0$ as $n \rightarrow \infty$, but $(f_n(z))_n$ pointwise…
Eddie
  • 201
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2 answers

Does the polynomial have n distinct real roots

Let $a_1,a_2,a_3...a_n$ be real numbers such that the polynomial $p(x)=x^n+a_1x^{n-1}+...+ a_{n-1}x+a_n$ has n distinct real roots. Does there exist $\epsilon$ >0 such that for all $b_1,b_2...b_n \in R$ with the property that $|a_j-b_j|<\epsilon$,…
a1bcdef
  • 2,265
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proving recursively defined sequence by induction

I would like to prove the following recursively defined sequence from $n-1$ to $n$ by induction. Im not realy sure about it. Any help or alternative ways to understand and prove it are highly appreciated : $0,1,4,12,35,98$ $a_0=0$, $a_1=1$,…
Mamba
  • 803
4
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2 answers

Derivative is a constant

A function $f:U\subset\mathbb{R}^n \rightarrow \mathbb{R}^m$, $U$ open, is differentiable in $p \in U$ if there exists a linear transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $f(p+v)=f(p)+T(v)+R(v)$, where $R(v)$ satisfies $lim…
Marra
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What functions satisfy such equation?

Let $f:(-a,a)\rightarrow \mathbb R$ be a continuous function such that $$ f(0)=\frac{f(-x)+f(x)}{2} \textrm{ for } |x|
user172903
  • 205
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1 answer

Showing $V_a^b\alpha = \sup \left\{\int_a^bfd\alpha:\|f\|_\infty\leq 1\right\}$

Let $\alpha:[a,b]\to\mathbb{R}$ be of bounded variation and right-continuous. Given $\varepsilon>0$ and a partition $P$ of $[a,b]$, construct $f\in C[a,b]$ with $\|f\|_\infty \leq 1$ such that $\int_a^bfd\alpha\geq V(\alpha,P)-\varepsilon$.…
Galois
  • 2,454
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2 answers

Cauchy sequence

Show that if $(x_{n})_{n}$ is a Cauchy sequence in X and $\lambda \in \mathbb{R}$, then the sequence $(\lambda x_{n})_{n}$, is also Cauchy in X. We know that for $(x_{n})_{n}$, we have $$\forall \epsilon >0:\exists N\in \mathbb{N} : n,m\ge N\implies…
dplanet
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Change of Variables Clarification

How can we show that $v(L(C)) = |\det DL|v(C)$ for any open cube $C$ an element of $\mathbb{R}^n$ and any linear transformation $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$, without direct applying the change of variables theorem? Thanks
James R.
  • 1,451
4
votes
1 answer

how to construct a polynomial

I have a question here: A finite sequence of real numbers $c_1, c_2, \dots, c_{n−1}$ is called saw– like if we have $(−1)^k(c_k − c_{k+1}) \leq 0$ for all $k = 1, \dots , n − 2$ or if we have $(−1)^k(c_k − c_{k+1}) \geq 0$ for all $k = 1, \dots , n…
benben112358
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  • 1
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eventually $\epsilon$-close sequences are bounded.

Let $ \epsilon $ >0. show that if $(a_n)_{n=1}$ and $(b_n)_{n=1}$ are eventually $\epsilon$-close sequences, then $(a_n)_{n=1}$ is bounded iff $(b_n)_{n=1}$ is bounded. proof; We can choose $M \geq 0$ such that $|a_i| \leq M$ for all $ i \geq…
user197848
4
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1 answer

Integration over a bounded set

Let $f,g: S\to \mathbb{R}$. If we assume $f$ and $g$ are integrable over $S$, then I'm trying to show: If $f$ and $g$ agree except on a set of measure zero, then $\int_S f=\int_S g$. Also, how can we show: If $f(x) \le g(x)$ for $x \in S$ and…
James R.
  • 1,451
4
votes
5 answers

Why for dirac function $\int_{-\infty}^{\infty}\delta(x)\ dx=1$

The dirac function is defined as $\delta(x)=\infty$ when $x=0$, $\delta(x)=0$ otherwise. I am wondering why we can derive $\int_{-\infty}^{\infty}\delta(x)\ dx=1$, or this is just a definition
89085731
  • 7,614
4
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2 answers

A question regarding a double series.

Let $\{a_{mn}\}$ be a double series, where $a_{mn}>0$ for all $m,n\in\Bbb{N}$. If $\sum\limits_{i=1}^\infty{a_{ik}}$ is finite for all $k\in\Bbb{N}$ and $\sum\limits_{j=1}^\infty{a_{hj}}$ is finite for all $j\in\Bbb{N}$, then…
algebraically_speaking
  • 41
4
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4 answers

How to prove that $\ln x\leq x-1 \forall x>0$?

I need to prove that $\ln x\leq x-1 \forall x>0$, using the Mean value theorem. For $x=1$, the equation is true. So, for starters I'll check for $x>1$. By applying the aforementioned theorem for $$f(t)=\ln t / [1,x]$$ we know that there is a…
Shevliaskovic
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