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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$f: \mathbb{R} \rightarrow \mathbb{R}, f(0)=1, f(x+y) \leq f(x)f(y)$. Show that if $f$ continuous in $x=0$, $f$ is continuous in $\mathbb{R}$.

So far I have shown that the equality only holds when $x=0$ or $y=0$. I also have found out that $f(nx) \leq f(x)^n$ for natural $n$ (otherwise the summation doesn't make sense). But I do not know how to deduce the continuity. My idea is to show it…
B.Swan
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"Constructions" with a countably infinite number of steps

Say I have a countably infinite initial set $A$ with $A \subset [0,1]$. My set satisfies the property that $\inf_{x\in A} |x-y| = 0,\,\forall{y\in [0,1]}$ (For example $A$ can be the set of rational numbers). I have a "construction" that works in…
zlog
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If $f\in C^1(\mathbb{R},M_n(\mathbb{R}))$ such that $f(0)=0$ and $f'(0)=I$, show that the image of f contains a regular matrix

If $f\in C^1(\mathbb{R},M_n(\mathbb{R}))$ such that $f(0)=0$ and $f'(0)=I$, show that the image of f contains a regular matrix. While trying to prove something (elementary) from representation theory, I came to a stop. This fact would complete the…
sonjcy
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Why $\lim\limits_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim\limits_{t\to 0}\frac{\sin t}{t}$?

Why $\displaystyle\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{t\to 0}\frac{\sin t}{t}$( and hence equals to $1$)? Any rigorous reason? (i.e. not just say by letting $t=x^2+y^2$.)
Eric
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Prove the following series converges

Let $u, v, w$ be real numbers such that $u+v+w=0$. Suppose that $\{b_k\colon k=0,1,2,\dots\}$ is a sequence of real numbers such that $\lim_{k\to\infty} b_k=0$. For $k=0,1,2,\dots$ define $a_{3k}=u b_k,$ $a_{3k+1}= v b_k,$ $a_{3k+2} = w b_k.$ Prove…
Thetexan
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A convergence problem in real non-negative sequence, $\sum_{n=1}^\infty a_n$.

We now have $a_n\geq 0$, $\forall n=1,2,...,$ and $\sum_{n=1}^\infty a_n <\infty$. Then I guess that $\lim_{n\to\infty} a_n \cdot n = 0$. But I realized that it is wrong. Since if we let $a_n = 1/n $ if $n = 2^i$ for some $i=1,2,...$ and $a_n = 0$…
Schrödinger's Cat
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Vector field and differential equation

We consider from $\mathbb{R}^2$ \ $\left \{ 0 \right \}$ the vector field $$X(x,y)=\left ( \dfrac{x}{x^2+y^2},\dfrac{y}{x^2+y^2} \right )$$ How to show that the differential equation $\dot{\gamma }(t)=X(\gamma (t))$ to any initial condition $p \in…
Melina
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Is my proof of $\lim_{x\rightarrow c}x^2=c^2$ correct?

I know the most common proof of $\lim_{x\rightarrow c}x^2=c^2$. But I wonder if my alternative proof is valid and correct. Here's my proof. Let $\varepsilon>0$, want to find a $\delta>0$ such that $\forall x\in\mathbb{R},0<|x-c|<\delta\Rightarrow…
Eric
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$\lim_{x\to\infty}(f(x)+f'(x))=0 \rightarrow \lim_{x\to\infty}f(x)=0$?

Suppose that $f:[0,\infty)\to\Bbb R$ is differentiable, and $\lim_{x\to\infty}(f(x)+f'(x))=0$. Prove that $\lim_{x\to\infty}f(x)=0$. I tried to show that $\lim_{x\to\infty}f(x)\neq0\rightarrow \lim_{x\to\infty}(f(x)+f'(x))\neq0$ If…
Arbitrary
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On the inner metric of loops in space

An injective Lipschitz map $f \colon S^{1} \to \mathbb{R}^n$produces a loop $M:=f(S^{1})$. The inner metric $d_M$ on $M$ is defined by $d_M(x,y):=\inf L(\gamma)$ over all curves $\gamma\subset M$ connecting $x$ with $y$. It is claimed…
MemeP
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Different ways to represent functions other than Laurent and Fourier series?

In the book "A Course of modern analysis", examples of expanding functions in terms of inverse factorials was given, I am not sure in today's math what subject would that come under but besides the followings : power series ( Taylor Series, Laurent…
jimjim
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Step functions and the characteristic function of rationals

A function $t: [a, b] \rightarrow \mathbb{R}$ is called a step function when a $k \in \mathbb{N}$ and numbers $z_0,...,z_k$ with $a = z_0 \leq z_1 \leq ... \leq z_k = b$ exist, such that for all $i \in \{1,2,...k\}$ the restriction $t…
ghshtalt
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Question about Sets with a Certain Property

I've been working on answering this question on and off for a while (months). I can't seem to solve it, and I've presented it to a few people who also cannot solve it. I will present a special case of the problem here, as I intuitively suspect that…
M10687
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Intuitive explanation for Jacobian matrix having max. rank

As part of a longer definition I came across the following: $f: X\subseteq\mathbb{R}^m \rightarrow \mathbb{R}^n$ ($X$ open, $m
ghostclicker
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Analysis: Showing a set can be bounded below in the reals

How do I go about showing the set $A=\left\{x^2+6x+6 : x\in\mathbb{R}\right\}$ is bounded below in the reals. Further,how may I go about finding the greatest lower bound for $A$? Thank you in advance
Taleri
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