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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smooth approximations of indicator function

How would I construct Schwartz functions $f_1^\epsilon$, $f_2^\epsilon$ on $\mathbb{R}$ such that $f_1^\epsilon(x)\leq\mathbb{1}_{[a,b]}(x)\leq f_2^\epsilon(x)$, and $f_1^\epsilon\rightarrow\mathbb{1}_{[a,b]}$,…
Kelly
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Monotone convergence theorem on $\ln(2)$ sum?

Given $\frac{1}{1+x}=\sum^{\infty}_{k=0}(1-x)x^{2k}$ for $x\in[0,1)$ I shall apply the 'monotone convergence theorem' on $[0,1)$ for calculating $\sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}$. In Wikipedia it's said that…
pinkpanther5
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Show that exists a constant $c>0$ such that: $\|f(x)\|\geq c\|x\| \forall x \in \mathbb{R^n}$

So , this is the exercise: Let $f$ be a linear transformation and injective in $\mathbb {R^n}\rightarrow\mathbb{R^m}$. For abuse , let's denote by $\|.\|$ the norm in both sides.Show that exists a constant $c>0$ such that: $$\|f(x)\|\geq c\|x\|…
HipsterMathematician
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Verifying that there is a piecewise linear function on a closed set that approximates it well

Question: Let $f$ be a continuous function on $[a,b]$. Show that there is a piecewise linear function $\phi$ on $[a,b]$ with $|f(x)-\phi(x)| < \varepsilon$ for $x \in [a,b]$. My proof: For any $\varepsilon > 0$ there is a $\delta >0$ such that…
emka
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Reduction to particular cases of a lemma

Lemma:Let $\phi(s)$ be a non-negative and non-decreasing function. Suposse that \begin{equation} \phi(r) \le C_1 \left[\Bigl(\dfrac{r}{R}\Bigr)^\alpha + \mu \right]\phi(R) + C_2 R^\beta] \end{equation} for all $r\le R \le R_0$, with…
user29999
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Given any function $g(x): \mathbb{R} \rightarrow \mathbb{R}$, find a function $f(x)$ such that $f(f(x)) = g(x)$.

This is a problem I was casually discussing with friends: Given any function $g(x): \mathbb{R} \rightarrow \mathbb{R}$, find a function $f(x): \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x)) = g(x)$. Is it possible to find a solution for…
nekodesu
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Continuity of $f\circ f \circ \cdots$

Let $f:D\to D$, where $D\subseteq \mathbb{R}^n$, be a continuous function. Under what conditions is $f\circ f \circ \cdots$ continuous? Here, $\circ$ stands for the composition operator and sometimes the notation $f^2=f\circ f$ is used. So in this…
Pantelis Sopasakis
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Proving linearity from differentiability

I dont know how to prove that if $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is differentiable and $f(x/2) = f(x)/2$ for each $x \in \mathbb{R}^n$, then $f$ is linear. Anyone could give me a hint?
Eduardo Silva
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$(f(1 - f(x)) = 1 - x^9$, $f(1) = 0$ and $f'(1) < 0$, then where is the real number $r$ such that $f(r) = r^{99}$?

If $f(1 - f(x)) = 1 - x^9$, $f$: R $\to$ R is differentiable, $f(1) = 0$ and $f'(1) < 0$, how to show there is a real number $r$ such that $$f(r) = r^{99}?$$ Edit: Taylor Theorem makes no use. I try to take $a = 1 - {1 \over n}$, where n is also a…
Yuki.F
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How to solve $y=\frac{(x-\sin x)}{ (1-\cos x)}$

I only see numerical approaches to solve this equation. Is there an analytical solution to solve $x$ as a function of $y$ for the range $(0,2 \pi)$? If there is no solution, is it possible to proof it?
varantir
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Infinite closed subset of $S^1$ such that the squaring map is a bijection?

Is there an infinite closed subset $X$ of the unit circle in $\mathbb C$ such that the squaring map induces a bijection from $X$ to itself?
Nishant
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Showing properties of discontinuous points of a strictly increasing function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be strictly monotonically increasing. (i) Is $f$ not continuous at $p \in \mathbb{R}$, there exists a non-empty, open interval $(a_p, b_p) \subset \mathbb{R}$ such that $f(x)\leq a_p$ for all $x < p$ and…
ghshtalt
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Example of nonlinear regular function with constant nonzero Jacobian

Can anyone give a nonlinear regular function from C^2 to C^2 with a constant nonzero Jacobian? It seems to me that the only such functions are linear. According to the Jacobian conjecture, a function from C^2 to C^2 with a constant nonzero Jacobian…
Craig Feinstein
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Is there anything wrong with this proposed proof of the irrationality of Euler's constant?

Let $\{\lambda_n\}$ be the sequence given by $H_n - \ln n$. We claim that $\lambda_n$ is irrational for every integer $n>1$ and justify this by the following argument: Assume that $\lambda_k$ is rational for some integer $k>1$ such that $H_k - \ln k…
user264948
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An Exercise on Inverse Function Theorem

Consider the mapping $f:R^2\rightarrow R^2$ given componentwise by: $f_1(x,y)=x+a_1x^2+2b_1xy+c_1y^2\\ f_2(x,y)=y+a_2x^2+2b_2xy+c_2y^2$ Determine a neighbourhood of $(0,0)$ as large as possible on which $f$ is invertible and bijective. Inverse…
iloveinna
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