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 $\{y_i\} \subset f([0,1])$ is increasing, there exists an monotonous $\{x_i\} \subset [0,1]$ such that $f(x_i)=y_i$?

Let $f \in C[0,1]$. If $\{y_i\} \subset f([0,1])$ is increasing, there exists an monotonous $\{x_i\} \subset [0,1]$ such that $f(x_i)=y_i$? Intuitively, from the function image, it is true. and how to prove that?
xunitc
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Prove that $\sup(aS)=a\sup (S)$

Let $S$ be a nonempty bounded set in $\mathbb{R}$ Let $a>0$ and let $aS=\{as: s\in \mathbb{R}\}$. Prove that $\sup(aS)=a\sup S$ I did my proof a different way then what the book did but I am not clear if my proof is still correct. The book proved…
user60887
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Is it the correct change?

Calculate the integral : $\displaystyle\iint_D\sqrt{x-y}\;dxdy\;,$ where $\;D=\big\{(x,y)\in\mathbb{R}^2\;\big|\; 1\leq y\leq4\;,\frac{4}{5}x\leq y\leq x\big\}\;.$ My attempt is : $\displaystyle\int_1^4\int_y^{\frac{5y}{4}}\sqrt{x-y}\;dxdy$ but I…
hi hi
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Double limit and repeated limit

If for a function a finite or infinite double limit $$ \lim\limits_{(x,y)\rightarrow(a,b)} f(x,y)$$ exists, and if for any $y \in Y$ there is a finite limit $$ \varphi(y) = \lim\limits_{x \rightarrow a}f(x,y) $$ then the repeated…
maplgebra
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What will Implicit Function Theorem be if Jacobian determinant is zero?

Implicit Function Theorem says if $F_i(x_1,..,x_n,y_1,..,y_n)=0,(i=1..n)$, and $\det\left(\frac{\partial F_i}{\partial x_j}\right)\neq0$, then $x_i$ can be expressed in terms of $\{y_j\}$. If $\det\left(\frac{\partial F_i}{\partial x_j}\right)=0$,…
1or2or3
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Maximum value of a Quadratic Function

I need help with the following problem: Find the maximum value of the quadratic function $f(x)=ax^2+bx+c$ on the interval $[0,1],$ where $|f(-1)| \leq 1, |f(0)| \leq 1, |f(1)| \leq 1.$ Thanks for all tips!
Mmath
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Debugging an analysis text

I'm having some problems understanding the following paragraph, which I read in a analysis script (hopefully I haven't made any translation errors): "A map $f:U \rightarrow Y$, where $U$ is open and $X,Y$ are Banach spaces, is continuous at $x' \in…
resu
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Proof of Cauchy Criterion

How did they get $|x_{n} - x_{n_{k}}| = \dfrac{\epsilon}{2}$ in the proof?
user79576
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Dense Countable Subsets of the Unit Disk

Consider a countable dense subset A of the unit disk D in R. Is it true that the sum over elements of A vanishes? Intuitively if we choose a neighborhood U of a point x in D we may find points in A arbitrarily close to x. Similarly we may do this…
Joseph Zambrano
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Asymptotics for an expression

I have a rather silly question, I guess. Suppose $(1+x)^{2y}\sim 1$ as $y\to\infty$. Does this imply $x\sim 1/y$ as $y\to\infty$?
Scuderi
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Proving a property of the Fourier transform

Let $0 < f ∈ L^1(\mathbb{R^2})$ almost everywhere. Prove: $|\hat{f}(x)| < \hat{f}(0) \quad \forall x \in \mathbb{R^d}\backslash\{0\}$. Could someone please help me with this question? I tried solving this with the step function but I don't seem to…
user14978538
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Proof on showing $\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$ for class $C^2$ function $f$

The task is as follows: Given: (a) function $f \in C^2$ (b) $f \geq 0$ and (c) $f'' \leq 0$ on $[a,b]$ Goal: Show $$\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$$ To get an understanding of the problem, I tried…
Cecile
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21 is a losing opening move in Sylver Coinage. Why??

Sylver Coinage Question. Is it just a losing opening move because the next player will respond with the prime 7? Does it relate to Hutchin's Theory? help. sylver coinage: Two players take turns naming positive integers, but an integer is off limits…
user404739
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double integrals and iterated integrals

Give an example (if any) for a non-integrable function $f:\mathbb{R\times R}$ $\to$ $\mathbb{R}$ with domain in $[0,1]^2$ such that both iterated integrals exists(i.e. in both order of integration). Here is what I have got: $$ f(x,y)…
user77440
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(non)-Benford's law?

Deriving stellar ages in galaxies its an intricate process that depends on many factors, such as the mathematical recipes of the tools one uses, the adopted stellar libraries and so on. Considering there are many available tools that provide quite…
faeriewhisper
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