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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The ratio of two strictly increasing functions

although it seems very simple and obvious, I have no idea how to give an analytical proof for this problem. I will be very happy if there are some smart ideas... Given, $f_1(a), f_2(a),..., f_n(a)$ and $g_1(a), g_2(a),..., g_n(a)$ are strictly…
Seyhmus Güngören
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Show the following are not connected in $\mathbb{R}^n$

Is the interior, boundary and closure of a connected set in $\mathbb{R}^n$ connected? I know the interior is not connected we can show it by a counterexample but I am not quite sure for the closure and boundary
a1bcdef
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Prove that $Df(p)=f(p)T$ where $T(q)=\int_{0}^1q$

Let $E=\mathcal{C}[0,1]$ provide with $\|\cdot\|_\infty$ norm. Let $f:E\to \mathbb{R}$, given by $f(p)=e^{\int_0^1 p}$. I need to prove that $f$ is differentiable. My approach: Let $p,q\in\mathcal{C}[0,1]$ then, $$f(p+tq)-f(p)=e^{\int_0^1…
Valent
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How to continue $f \in C ^\infty[a;b]$ to $f \in C ^\infty (\mathbb {R })$?

Is there a general way to extend a smooth function on a closed interval $[a,b]$ to one that is defined on the entire $\mathbb R$? It is not OK to reflect this function in points $a$ and $b$, and then in points $a -(b-a) $ and $b +(b-a) $ etc…
user228494
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Finding all accumulation points of the a set.

I am working on homework for my intro to analysis class and I was assigned a problem to find all accumulation points of the set $S=\{x\mid x\in[0,1]$ and $x$ is rational$\}$. I hodge podged a solution but I feel like it may be incorrect. If anyone…
EvHi
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Connections between Cesaro summation and Borel summation of series

Let $\sum_{n=0}^\infty x_n$ be a given series of numbers, let $S_n=\sum_{k=0}^n x_k$, $n=0,1,2,...$, let $g\in \mathbb R$. We say that this series is convergent to $g$ in the sense of Cesaro if $$ \frac{S_0+S_1+..+S_n}{n+1}\rightarrow g $$ as…
Alex
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Boundedness of an integral

Got stuck on this one: Show that there is a constant $C>0$ such that $$\left|\int_0^x \frac{\sin (N+1/2)t}{\sin t/2} dt \right|\le C$$ for all $x\in [-\pi,\pi]$ and integer $N\ge 1$. I thought this should follow from $\int_0^\infty \frac{\sin…
TCL
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Technical issue - how does author substitute in these integrals?

A little prerequisite knowledge first, the author defines a 'mollifying' function $\rho_k: \mathbb{R}^n \rightarrow \mathbb{R}$ to have several properties, but that aren't relevant to my question (from what I understand, it exists to smooth…
Necroticka
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Switching Limits and Suprema

suppose, $f(x,y)$ is a bounded continuous function on $\mathbb{R}^2$. Consider $$\lim_{ y \rightarrow y' }\> \sup_{x \in \mathbb{R}} f(x,y).$$ In how far can you switch suprema and limits, or which possible term comes closest to switching these…
shuhalo
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How to find the limit properly?

I would like to find or solve the limit of: $$\lim_{n \to \infty} \frac{80^{(n+1)/4}}{37^{(2n+3)/4}}$$ My idea was somehow non-intuitive: $$\lim_{n \to \infty} \frac{80^{(n+1)/4}}{37^{(2n+3)/4}} \leq \lim_{n \to \infty} \frac{3^{4(n+1)/4}}…
Mamba
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Uniform convergence and composition

Suppose we have a sequence of analytic functions $f_j:(1,M) \to (1,M)$ for some $M \in \mathbb{R}^+$. We are guaranteed that there is an $n \in \mathbb{N}$ such that its $n$'th iterate $$g_j(x) =f_j( f_j(...(n\,times)...(f_j(x)) = (f_j \circ f_j…
JmsNxn
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If $\int_1^x f(t)^2dt \le \frac{x^3-1}{3}$ then $\int_1^2 f(t)dt \le \frac{3}{2}$

If $f:[1,2]\to [0, \infty )$ is an Riemann integrable function such that $\int_1^x f(t)^2dt \le \frac{x^3-1}{3} , \forall x \in [1,2]$. Prove that $\int_1^2 f(t)dt \le \frac{3}{2}$ . First, I used Cauchy's inequality: $(x-1) \int_1^x f^2(t)dt \ge…
npatrat
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Show that if $U \subseteq C$ and $V \subseteq C$ are both open and convex sets, then the set $U \cap V \subseteq C$ is open and convex as well.

I think you have to prove that as it is the intersection then both are in open and convex sets seeing as they are on their own. Don't really know how to put this down in notation though.
Tom
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Prove that function with positive second derivative grows faster than a linear function

The statement of the problem is as follows, Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and $f''(0)$ exists and is greater than $0$. We also have $f(0) = 0$. Prove that there exists an $x > 0$ such that $f(2x) > 2f(x)$. I have…
Peter Booth
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Regularity of the distance function

Let $\Omega \subseteq \mathbb{R}^N$ be open and bounded, and set: $$d(x):=\text{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|\;.$$ I would appreciate if somebody could verify my proof. I tried to show that it is Lipschitz…
user16847
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