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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$ \int_{\{u>j\}} (u-j) dx = \int_j^\infty | \{u>j\}| dt?$

I have seen that if $u$ is a summable function (in fact, I saw that if $u \in W^{1,p}$, but I think that summable is sufficient) in $\mathbb{R}^n$ then \begin{equation} \int_{\{u>j\}} (u-j) dx = \int_j^\infty | \{u>j\}| dt \end{equation} where $j$…
user29999
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Proving a combination of differentiability

$f:\mathbb{R}\to \mathbb{R} $where$$f(x) = \begin{cases} \dfrac{P(x)}{x^n}e^{-1/x^2}& \text{if $x\ne 0$}, \\ 0 &\text{if $x = 0$}.\end{cases}$$ Where P(x) is a polynomial and $n\geq 0$ is an integer. Prove that $f$ is differentiable everywhere and…
user113494
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Prove that there is at least one number from $[a;b] : f(c)=g(c)$

$f(x)$ and $g(x)$ are two continuous functions at $[a;b]$ and for every number $x$ from $[a;b]$ there is at least one number $y$ from $[a;b] / f(x)=f(y)$ Prove that there is at least one number from $[a;b] : f(c)=g(c)$
Fertouch
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Show that if $f(1)=1$, then there exists a constant $\alpha$ such that $f(x)=x^\alpha$ for all $x \in (0, +\infty)$.

Let $f: (0, +\infty) \to\mathbb R$ be a differentiable function such that $f(xy)=f(x)f(y)$ for all $x,y \in (0, +\infty)$. Show that if $f(1)=1$, then there exists a constant $\alpha$ such that $f(x)=x^\alpha$ for all $x \in (0, +\infty)$. So…
user113009
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How prove $f=0$,if $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}=f$

Question: let $D$ is Bounded closed region,and Assume $f$ is Continuous differentiable on $D$,if such $$\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial y}+\dfrac{\partial f}{\partial z}=f,f|_{\partial D}=0$$ show that $f=0$ this problem…
user94270
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Show that u is a limit point for the set.

Let $a_1, a_2, a_3$,.... be a sequence so that $a_1
user102839
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How prove this function have bounded and such $\lim_{r\to\infty}\frac{\int_{-r}^{r}|f(x)|dx}{2r}=k$

Question: Let $f$ is continuous on $R$,and such $$\lim_{r\to+\infty}\dfrac{\int_{-r}^{r}|f(x)|dx}{2r}=k$$ where $k$ Can be infinite. prove or disprove $f(x)$ is Bounded function? My try: I think we can consider follow two case: case 1: if…
user94270
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Prove that $f(x)=x^\alpha$ for all $x>0$.

Suppose that $f:(0,\infty) \to \Bbb R$ is differentiable and that there exists a constant $\alpha$ belonging to $\Bbb R$ such that $x*f'(x)=\alpha*f(x)$ for all $x>0$ and $f(1)=1$. Prove that $f(x)=x^\alpha$ for all $x>0$. I tried integrating both…
user113009
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Prove that $f(x)=x$ for all $x\geq 0$

Suppose that $f$ is continuous on $[0,\infty)$ such that $f(x)>0$ for all $x>0$ and that $f^2(x)=2\int_0^x f(t)\,dt$ for all $x>0$. Prove that $f(x)=x$ for all $x\geq 0$. Attempt at a proof: Let $f(t)=t$ for all $t\geq 0$. Since $f$ is continuous…
user113009
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Do closed intervals exist?

If 0.999.. = 1, does that mean that infinitesimals are not allowed in $(-\infty,1)$? Otherwise, we would have $0.9 \in (-\infty,1)$, $0.99 \in (-\infty,1)$, $0.999\in(-\infty,1)$, ad infinitum.
Julian Cienfuegos
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How prove this only point such $f(x,y)$ obtain the maximum

Question: let $$D=\{u=(x,y)\in R^2\colon||u||=\sqrt{x^2+y^2}\le\dfrac{1}{2}\}$$ and $f(u)=f(x,y)$ is all plane continuously differentiable,and such $$||\nabla f(0,0)||=1,||\nabla f(u)-\nabla f(v)||\le||u-v||$$ let $\forall u,v\in D$, show…
math110
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Show that V must be finite dimensional

Let $V$ be a normed space in which every bounded sequence has a convergent subsequence. Show that $V$ must be complete. Show further that $V$ must be finite-dimensional. I've done the first part in showing $V$ is complete so I've got a grasp on…
Raul
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$\varepsilon$-$\delta$ proof of $\sqrt{x+1}$

need to prove that $ \lim_{x\rightarrow 0 } \sqrt{1+x} = 1 $ proof of that is: need to find a delta such that $ 0 < |x-1| < \delta \Rightarrow 1-\epsilon < \sqrt{x+1} < \epsilon + 1 $ if we choose $ \delta = (\epsilon + 1)^2 -2 $ and consider $…
DH.
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Proof inequality with log

How can I prove that there exist $n_0$, $c$ such that for all $n>n_0$: $$n^{\log_2{n}}\le c2^{n}$$ (So I mean the log of n with base 2). Can anybody help me?
Roos Jansen
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integral of a continuous function bounded. Prove function is identically zero.

Let ${\rm f}:{\mathbb R}^{k} \to {\mathbb R}$ be a continuous function. Assume that for any $a > 0$ and any $k$-cell $Q_{a}$ of side length $a$ $\left(~\mbox{and therefore volume}\ a^{k}\right)$ we have $\displaystyle{% \left\vert\,\int_Q {\rm…
user2496006
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