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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Give an example of a sequence of continuous functions which converges on a compact set to a function that has an infinite number of discontinuities.

Give an example of a sequence of continuous functions which converges on a compact set to a function that has an infinite number of discontinuities. Analysis is something that is very difficult for me, and I am not fully sure what I am supposed to…
user99930
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Infinite gradient and continuity

Would I be right in thinking that although the function $f(x)=x^2\sin({1\over x^2})$ has infinite gradient, it still uniformly continuous? Thanks.
jose
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Prove that for a polynomial function $f$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$

Prove that for a polynomial function $f$ with degree $n$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$. Give me some hints for this and please explain to me how you have come to those hints.
le4m
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An inequality of $\int_0^1 |f(x)|dx$

Question: If $f\in C^1[0,1]$, show that $$\int_0^1 |f(x)|dx\le\max\left\{\int_0^1 |f'(x)|\,dx,\;\bigg|\int_0^1 f(x)\,dx\bigg|\right\}.$$ I have tried to make connection between $|f|$ and $|f'|$ by using $$(tf(t))'=f(t)+tf'(t),$$ integrate the…
NGY
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I'm stuck with this.. (number 9,6 and 3)

Hello guys/girls I was bored and I just played around with math. I am stuck and it's about raised numbers. (9, 6 and 3) So this is how you calculate it. (same method for all numbers) Raise 3, 6 and 9 each from 1 to 10. If the product is more than…
Luce newlen
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French metro metric: difficulty to prove that $d(x, y) = 0\iff x = y$.

I think that it is related to the special definition of the metric in my book: $$d(x, y) = \begin{cases}||x - y||,\mbox{ if }\exists \alpha\in\mathbb{R}: \alpha x + (1-\alpha) y = 0;\\ ||x|| + ||y||, \mbox{ otherwise.}\end{cases}$$ This way, for $x…
Igor Deruga
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Updated: Constructing a bijection between $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\mathbb{R}$

I am supposed to construct a bijective function for the interval: \begin{align} I_2=\left(-\frac{\pi}{2} ,\frac{\pi}{2} \right] \longrightarrow \mathbb{R} \tag{Problem} \end{align} I first tried the easier case,…
Spaced
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On the existence of $\sqrt{2}$ (guided)

Introduction: This is a homework assignment of mine, first I want to mention that I am aware of that there are many proofs all over the internet (including this site) about the existence of $\sqrt{2}$. However, in my assignment I am somewhat…
Spaced
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The projection on the first factor is a bijection

I require some clarification and hints on the following Problem: \begin{align}f: X \longrightarrow Y \end{align} The image $p$ is defined as: \begin{align} p: G_f &\longrightarrow X \\ (x,y) &\longmapsto x\end{align} With $G_f=\lbrace (x,y) \in X…
Spaced
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Find every bijection of non-zero real numbers whose inverse is its reciprocal

Find all functions $f $: $\mathbb R^{×}$ $\to$ $\mathbb R^{×}$ that are one-to-one and onto and such that $f^{−1}(x)= 1/ f(x)$, Where $\mathbb R^{×}=\mathbb R-${0} My approach:= At first I have consider $f(x)=x$ and $f(x)=1/x$ then this does not…
SUJAN DAS
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With the epsilon-delta definition of continuity, are the rationals continuous?

My current knowledge is that a function is continuous at a point $x=a$ if and only if, for any $\epsilon>0$, there exists some $\delta>0$ such that $$ |x-a|<\delta \implies |f(x)-f(a)|<\varepsilon. $$ If I have a function $f: \mathbb{Q} \to…
Samuel Han
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How to prove the implicit function theorem fails

Define $$F(x,y,u,v)= 3x^2-y^2+u^2+4uv+v^2$$ $$G(x,y,u,v)=x^2-y^2+2uv$$ Show that there is no open set in the $(u,v)$ plane such that $(F,G)=(0,0)$ defines $x$ and $y$ in terms of $u$ and $v$. If (F,G) is equal to say (9,-3) you can just apply the…
user9352
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How prove this Mathematical Analysis by Zorich, from the chapter on continuous functions.

Let $P_n$ be a polynomial of degree $n$. For a function $f:[a,b]\to\mathbb{R}$, Let $\Delta(P_n) = \sup_{x\in[a,b]} |f(x)-P_n(x)|$. and $E_n(f) = \inf_{P_n} \Delta(P_n)$. A polynomial $P_n$ is the best approximation of degree $n$ of $f$ is…
math110
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A question about consistency

These Definitions come from Analysis I textbook of Tao, My question is: what is mean of 'Definition 6.1.2 is consistent with Definition 4.3.4',and why is it so clear that they are consistent. How do we verify this 'consistency'? Thank you!
Andrew Li
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n-th derivative condition to become polynomial for real function

I've seen a similar result in complex analysis, when an entire complex function satisfies $f(z)f^{(n)}(z)=0$ for all $z \in \mathbb{C}$ implies that $f(z)$ is polynomial. What if when $f: \mathbb{R} \rightarrow \mathbb{R}$ is a $n$ times…
Ajat Adriansyah
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