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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If $a,b \in \Bbb R$, prove that $|ab| \le (a^2+b^2)/2$

So far I have the first case when $a=b$: \begin{align*} |ab| &= |b^2|\\ &=|b|^2\\ &=\frac{2|b|^2}2\\ &=\frac{b^2+b^2}2\\ &=\frac{a^2+b^2}2 \end{align*} Case 2: $a>b$ Case 3 $a
david stocker
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inequality of some integrals of continuously differential function.

Let $f:[a,b]$→$\mathbb{R}$ be a continuously differential fuction satisfying f(a)=0. My goal is to show that $$\int_{a}^{b} |f(x)|^2 dx \le \frac{(b-a)^2}{2} \int_{a}^{b} |f'(x)|^2 dx $$ My attempt is following: By FTC, $f(x)=f(a)+\int_{a}^{x}…
NNNN
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Solution to a tricky inequality (math analysis)

Let $p>1$ and put $q=\frac{p}{p-1}$, so $1/p+1/q=1$. Show that for any $x>0$ and $y>0$, we have $$ xy \le \frac{x^p}{p}+\frac{y^q}{q}$$ And find where the equality holds. So far, I have simply tried to multiply through the RHS of the above…
johnsteck
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How prove this analysis function $a\le\frac{1}{2}$

let $$f(x)=\begin{cases} x\sin{\dfrac{1}{x}}&x\neq 0\\ 0&x=0 \end{cases}$$ show that:there exsit $M>0,(x^2+y^2\neq 0)$ , $$F(x,y)=\dfrac{f(x)-f(y)}{|x-y|^{a}}|\le M \Longleftrightarrow a\le\dfrac{1}{2}$$ My try: (1)if $a\le\dfrac{1}{2}$,…
math110
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How to show this identity involving multi-index?

I need some help for showing, $$\displaystyle \sum_{\beta \leq \alpha} \binom{\alpha}{\beta}(-1)^{|\alpha-\beta|}=0,$$ where $\alpha, \beta\in\mathbb N_0^n$ and $N_0=\mathbb N\cup \{0\}$. Any help will be welcome, thanks..
PtF
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Derivative $f : GL_n(\mathbb{R}) \to GL_n(\mathbb{R}) : A \mapsto A^{-1}$

Given a continuous $f : GL_n(\mathbb{R}) \to GL_n(\mathbb{R}) : A \mapsto A^{-1}$, I want to show that for $g \in GL_n(\mathbb{R})$ the derivative $Df(g)$ equals $$ H \mapsto -g^{-1}Hg^{-1} $$ for $H \in Mat(n, \mathbb{R})$. I tried proving…
Math Student 020
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Surjectivity of a function considering $f \circ h=Id_Y$

\begin{align} f: \mathbb{R}^2 &\longrightarrow \mathbb{R} \\ (x,y) & \longmapsto x+y \end{align} Question: Is this function surjective? It seems clear to me that this function must be surjective, because the point $(x,y) \in \mathbb{R}^2$ maps to…
Spaced
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Need help showing supremum does not exist

(i) S is a subset of R and β ∈ R. Let T = {s + β : s ∈ S }. Show that if Sup S exists, then so does sup T. Moreover, Sup T = β + Sup S. (ii) Using what you have proved in (i), show that if x ∈ R, x ≠ 0 and S = {gx : g ∈ Z }, then Sup S does not…
Juniper Leaf
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The set $A=\{(x, y)\in \mathbb{R}^2:|x|=|y|\}$ is connected

Two disjoint sets $A$ and $B$, neither empty, are said to be mutually separated if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is called connected if it is not disconnected. With…
Roiner Segura Cubero
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An interesting question on function iteration

$f$ is a continuous, strictly increasing function and $f(x+1) = f(x)+1$ We can know that $$\forall x \in \mathbb{R},\lim_{n\rightarrow + \infty} \frac{f^{n}(x)}{n}$$ exists, and its value doesn't rely on x Is this proposition right?(Probably…
Dragon_math
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Is there a non trivial smooth function that has uncountably many roots?

(on a bounded domain). I believe that such a function could not exist since every $C^\infty$ function can be approximated by a sequence of polynomials and every polynomial has a finite number of roots so it would not be possible for something that…
user847197
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If $a_n=\sin(n^2)$, how to construct (if possible) a subsequence such that $\lim a_{n_k}\rightarrow 0$?

For $a_n=\sin(n)$, it can use the continued fraction method to construct the subsequence, so that $\lim a_{n_k}\rightarrow 0$. But for $a_n=\sin(n^2)$, how to construct (if possible) a subsequence such that it converges to $0$ ?
MathFail
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Derivative at $0$ of $\int_0^x \sin \frac{1}{t} dt$

Let $f(x)=\int_0^x \sin \frac{1}{t} dt \textrm{ for } x \in \mathbb R$. Is $f$ differentiable at $0$ ?
user111
  • 447
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Extending a continuous function defined on a subset of $\mathbb{R}$

Let $E$ be a subset of $\mathbb{R}$ and let $f$ be a continuous function defined on $E$. Is it true that $f$ can always be extended to a function $\tilde{f}$ defined on $\mathbb{R}$, which is still continuous on $E$? I know that we cannot ask…
Jiu
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$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

On $[a,b]$ function $f$ is differentiable for arbitrary order, and $f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$. I first try Taylor expansion at $x_0=\frac{a+b}{2}$, and drop higher order term ($(x-x_0)^3$ terms). But this…
MathFail
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