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
4
votes
0 answers

Why Characterize Structure Preserving Functions in Terms of Pre-images?

In Analysis, functions are often characterized as "structure-preserving" if structures from codomains are preserved into the domain under the preimage operation. Specifically, if we let $f: (A, S_1) \rightarrow (B, S_2)$ be a function s.t. $S_1…
user1770201
  • 5,195
  • 6
  • 39
  • 71
4
votes
2 answers

Infinite series for $e$...

How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...
user368484
4
votes
1 answer

Proof that $f'(x) = g'(x)$ if and only if $f = g + c$

Good evening, I need to prove that $f'(x) = g'(x)$ for all $x \in (a,b)$ if and only if there exist a $c \in \mathbb{R}$ with $f = g+c$. We know that $f:[a,b] \rightarrow \mathbb{R}$ and $g:[a,b] \rightarrow \mathbb{R}$ are continuous and…
Just a Student
  • 157
  • 2
  • 8
4
votes
1 answer

When can one pass a linear operator under the integral?

Specifically, one the webpage: http://mathworld.wolfram.com/GreensFunction.html It is written that $$ \int\mathcal{L}G(x,s)f(s)ds = \mathcal{L}\left(\int G(x,s)f(s)ds\right) $$ where $G$ is a Green function. Why can the linear operator be pulled…
Quoka
  • 3,133
  • 9
  • 22
4
votes
1 answer

A smooth path passing through infinitely many points of a given sequence

While doing my research I came across an interesting question. Let $\{\bf{x}_n \}\subset R^m$ be any sequence of points with $\bf{x}_n\rightarrow \bf{0}$, as $n \rightarrow \infty$. Is there a smooth path $c: [0,a) \rightarrow \mathbf{R}^m$, $a>0$,…
Mohammadreza Bidar
  • 209
4
votes
2 answers

Question about integral and unit step function

The unit step function $I$ is defined by $$ I(x)= \begin{cases}0,\quad x \le 0, \\ 1,\quad x>0. \end{cases} $$ Let $f$ be continuous on $[a,b]$ and suppose $c_n\geq 0$ for $n=1, 2, 3,\ldots$ and $\sum_n c_n$ is convergent. Let $\alpha=\sum_{n=1}^{N}…
niagara
  • 735
4
votes
1 answer

If $\sum a_n$ converges, and if {$b_n$} is monotonic and bounded, prove $\sum a_n b_n$ converge.

I proved this way: Since {$b_n$} is bounded, let $|b_n| \le \alpha$ and $\alpha$ is an upper bound. Since $\sum a_n$ converges, there exists N such that $|\sum_{k=m}^{n} a_k| \le \frac{\epsilon}{\alpha}$ for every N $\le$ n,m. Then…
niagara
  • 735
4
votes
2 answers

Showing a function has only one point of continuity.

Let $$f(x) = \begin{cases}\;\;\, x\;\;,\;\text{ if } x \in \mathbb{Q}\\ -x\;\;,\; \text{ if } x \in \mathbb{R}\setminus \mathbb{Q} \end{cases}$$ (i) Determine the point or points of continuity of $f$. (ii) Show that the point of points of…
emka
  • 6,494
4
votes
3 answers

Prove that any continuous integer-valued function of a real variable is constant.

I'm stuck on a question which asks to prove that any continuous integer-valued function of a real variable is constant. In my lecture notes we are told that a map is continuous if the preimage of any open set is open. How do I use this to answer the…
Denis
  • 241
4
votes
2 answers

Showing that the evaluation map is not continuous?

Earlier today I was reading over some analysis notes, and I noticed something interesting and unintuitive. The metric $d_1: \mathcal{C}([0,1],\mathbb{C}) \times \mathcal{C}([0,1],\mathbb{C}) \to \mathbb{R}$ was defined via $d_1(f,g) =…
Charlie Fairfax
  • 43
4
votes
3 answers

For what values of $a$ will the following sequence converge?

$a \in \mathbb R$ has the decimal expansion $a = a_0.a_1a_2a_3 \ldots a_n \ldots$ Find all values for $a$ for which the sequence $\{a_n\}_{n=1}^{\infty}$ converges. I rule out irrationals first, because if they can't be represented with integer…
pad
  • 3,017
4
votes
2 answers

Problem with partition of unity in the Borel theorem

The Borel theorem says that for arbitrary sequence $(f_n)_{n=0}^\infty$ of smooth functions $f_n : \mathbb R\rightarrow \mathbb R$ with compact supports there exists a smooth function $F: \mathbb R^2 \rightarrow \mathbb R$ such that…
Richard
  • 4,432
4
votes
3 answers

How can I prove the monotonicity of a function?

Given is the function $$f(x) =\frac{x}{\sqrt{1-x^2}}$$ How can I prove that this function is monotonic and thus injective?
Marco7757
  • 143
4
votes
3 answers

Power series of a function

I am wondering if there are any functions $f(x)$ such that it cannot be expressed as a power series of $x$? This might turn out to be a silly question, but I can't think of one at the moment. Thanks!
Nikolaj Goodwin
  • 43
4
votes
1 answer

Showing the Cantor function is not Lipschitz.

This is one I am having a lot of difficulty with. I'm not sure how to show that the Cantor function (or 'Devil's Staircase) is not Lipschitz.
emka
  • 6,494
1 2 3
…
99 100