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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Function is constant given condition on integral of it's product with a member in a family of functions

Let $S=\{g(x)|g(x) $ is integrable in $[a,b]$ and $\int_a^bg(x)dx=0\}$,$f(x)$ is continuous on $[a,b]$.if $\int_a^bf(x)g(x)dx=0$ for every $g(x) \in S$，then $f(x)$ is constant I tried in several direction but seemingly can't solve it genuinely. I…
math
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All real numbers can be expressed as a limit of rational numbers?

RTP Let $C$ be a set of Cauchy sequences. $\forall x \in {\Bbb R}, \exists \{a_n\} \in C$ sucht that ${a_n} \to x$. I have no clue to even start this problem. All I know so far is that $\Bbb R$ is a set of equivalent classes of the limit of…
hyg17
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Locally Lipschitz and polynomial map

I'm trying to prove that: if $p:\mathbb{R}^p \to \mathbb{R}^q$ is a polynomial map then $p$ is Lipschitz over compacts.
jon jones
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Ask a step in Proof of L'Hospitals Rule

Suppose that $f$ and $g$ are differentiable functions on an open interval $I$ and that $p \in I$. If $\lim_{x \to p} f(x) = \lim_{x \to p} g(x) = 0$ and if $$\lim_{x \to p} \frac{f'(x)}{g'(x)}$$ exists and equals a real number $l$ then $$\lim_{x \to…
Mariana
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Ask the proof of $f(x)=x^2$ is not uniformly continuous on $[0, \infty)$

$f(x)=x^2$ is not uniformly continuous on $[0, \infty)$. Consider the function $f(x)=x^2$. Fix a point $P \in \mathbb{R}, P>0$ and let $\epsilon >0$. In order to guarantee that $|f(x)-f(P)|<\epsilon$, we must have (for x>0) $|x^2-P^2|<\epsilon$,…
Mariana
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A question on the limit of Thomae function

The definition of Thomae function is $$f(x) = \begin{cases} \dfrac{1}{q}, & \text{if $x=\dfrac{p}{q}$} \\[2ex] 0, & \text{if $x$ is irrational} \end{cases}$$ where $p\in \mathbb{Z}, q\in \mathbb{Z}^+$ and $\gcd(p,q)=1.$ The conclusion is $\lim_{x…
Mariana
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estimate on a function with compact support

let $U\subseteq\mathbb{R}^n$ be open and $f\in C_c^2(U)$ satisfy $f\geq0$ in $U$. I want to get an estimate of the function which is $$\sup_U\frac{|Df|^2}{f}\leq2\sup_U|D^2f|$$
Emiya
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Prove that the directional derivative is the inner product of the direction and the gradient

I want to prove Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is differentiable, and let $v$ be a unit vector in $\mathbb{R}^n$. Prove that the directional derivative of $f$ at point $\bar{x}$ with respect to direction $v$…
Scanners
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Find continuous function $f$ such that $\int_0^{\infty} f(x) \left(\frac{x^2}{1+x^2} \right)^n dx =\infty$ for all integers $n \geq1$.

Let $f(x)$ be a continuous function on $[0,\infty)$. I want to find $f$ such that $\int_0^{\infty} f(x) \left(\frac{x^2}{1+x^2} \right)^n dx =\infty$ for all integers $n \geq1$. Let $g_n(x) = \left(\frac{x^2}{1+x^2} \right)^n$, then I know $g_n(x)…
phy_math
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The graph of $f$

My question is kind of silly but I’m wondering why in my textbook, their is a drawing of a function $f$ whose graph continues after the limit point into the $x_0 +\delta$ region of the interval. I don’t understand why this is so. I think about…
Fredsaffiyre
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Show that $\lim\limits_{x\to\infty}f(x)=0$ if $f:[0,\infty)\to\mathbb{R}$ is nonnegative, integrable, and uniformly continuous.

I have several questions here. First, does the fact that $f$ is integral means the integral is finite? I was wondering if I can prove that $L(f,P)$ goes to infinity for some partition $P$ if $\lim\limits_{x\to\infty}f(x)\neq0$. Secondly, I think I…
WinnieXi
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Does a continuous map on a compact subset of $\mathbb{R}^n$ always have a compactly supported extension to $\mathbb{R}^n$?

Let $K$ be a compact subset of $\mathbb R^n$ and $f : K \to \mathbb R^p$ a continuous map. Is there always an extension $\tilde f$ of $f$ to $\mathbb R^n$ as a whole such that $\tilde f$ is continuous and compactly supported?
KCJV
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Can anyone help me show that $ \sum_{n=2}^{\infty} \frac{\cos(\pi n)}{\log(\log(n))}$ isn't absolutely convergent?

I have to show that $\displaystyle \sum_{n=2}^{\infty} \frac{\cos(\pi n)}{\log(\log(n))}$ isn't absolutely convergent. I thought about doing the following: $\begin{align}\displaystyle\sum_{n=2}^{\infty}\left|\frac{\cos(\pi…
user926600
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