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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An Simple Analysis Problem

$\varphi:\textrm{R}\rightarrow\textrm{R}$ is continuous, $\lim_{x\rightarrow\infty}\varphi(x)-x=\infty$, and $\{x\in\textrm{R}|\varphi(x)=x\}$ is a finite non-empty set. If $f:\textrm{R}\rightarrow\textrm{R}$ is continuous and $f\circ\varphi=f$,…
Starlight237
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Rudin Theorem 2.47 - Connected Sets in $\mathbb{R}$

I need help with the proof of the converse, as given by Rudin in Principles of Mathematical Analysis, to the following theorem: Theorem 2.47: A subset $E$ of the real line $\mathbb{R}^1$ is connected if and only if it has the following property: If…
user70962
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I want to study $\sqrt[n]{n}$ and its behavior.

As I was studying some limit problems, I came across $$\sqrt[n]{n}$$ and astoundingly found out that the graph of this has a maximum when $n = e$. I thought there is no way that this is not a famous fact and I am very interested in it. I looked up…
hyg17
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Show that the equation $\log(1+e^x)=\cos(x)$ has infinitely many negative solutions.

Show that the equation $\log(1+e^x)=\cos(x)$ has infinitely many negative solutions. Find out if there is a positive solution and if it is unique. From the graph I can see that it has an infinitely many negative solutions and that it has one…
user4167
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Construction of a join function of infinitely many derivatives.

I am curious if anyone can construct a function made up of more than one e.g. $|x| = x, x\geq 0$ and $-x, x\leq 0$. However I would require that it must be infinitely differentiable and in the above case of $|x|$ it must be infinitely differentiable…
user61038
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Class $C^{- \infty }$ functions?

If my understanding is correct, a class $C^{-1}$ function (in terms of smoothness, of course) can be thought of as a function which integrates to a class $C^{0}$ function. And when we differentiate (in the appropriate sense, of course) it, we can…
lel
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Constructing a continuous function whose graph seems 'special'

I've been reading through Zorich's "Analysis I" book recently, and I came across this nice little exercise. Let $f: [0,1]\to \mathbb R$ be a continuous function such that $f(0)=f(1)$. Show that for any $n\in \mathbb N$ there exists a horizontal…
Sam
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Bounded partial derivatives imply continuity

As stated in my notes: Remark: Suppose $f: E \to \mathbb{R}$, $E \subseteq \mathbb{R}^n$, and $p \in E$. Also, suppose that $D_if$ exists in some neighborhood of $p$, say, $N(p, h)$ where $h>0$. If all partial derivatives of $f$ are bounded, then…
JamieUSF
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Mean Value theorem and Newton's Method

I am trying to prove that: given $x_0, x_1, x_2 \ldots$ the sequence of approximations to $\pi$, use the mean value theorem to show that $|\pi-x_{j+1}| = |\tan c_j||\pi - x_j|$, where $c_j$ is some number between $x_j$ and $\pi$. So I do the…
user38268
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calculate $\lim_{x \rightarrow 0}\left ( x^{-6}\cdot (1-\cos(x)^{\sin(x)})^2 \right )$

as in topic, my task is to calculate $$\lim_{x \rightarrow 0}\left ( x^{-6}\cdot (1-\cos x^{\sin x})^2 \right )$$ I do the following: (assuming that de'Hospital is pointless here, as it seems to be) I write Taylor series for $\sin x, \cos x$ $$\sin…
fdhd
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Why is arctangent smooth?

I just read a proof by Spivak that $\arctan(x)$ has the $(2n+1)$-th Taylor polynomial at zero $$x - \frac{x^3}{3} + \cdots + (-1)^n\frac{x^{2n+1}}{2n+1}$$ The proof relied on the assumption that $\arctan(x)$ has $2n +1$ derivatives in order for the…
Chris
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total variation of a measure

I ran across this question in my analysis textbook. I just cannot prove this. Suppose $\mu$ is a complex measure on $X$ such that $\mu(X)=1$ and $|\mu|(X)=1$. Show $\mu$ is a positive real measure.
john
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Infinitely valued functions

Is it possible to define a multiple integral or multiple sums to infinite order ? Something like $\int\int\int\int\cdots$ where there are infinite number of integrals or $\sum\sum\sum\sum\cdots$ . Does infinite valued functions exist (Something like…
naanwa
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Prove that $h'(t)=\int_{a}^{b}\frac{\partial\phi}{\partial t}ds$.

Let $\phi:[a,b]\times[c,b]\to \mathbb{R}$ be continuous. Define $h:[c,d]\to \mathbb{R},h(t)=\int_{a}^{b}\phi(s,t)ds$. Assume that $\frac{\partial\phi}{\partial t}$ exists and is continuous on $[a, b]\times[c, b]$. Prove that…
Mathematics
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Application of the Intermediate Value Theorem

Suppose $f: [1,2] \to [5,7]$ is continuous. Show that $f(c)=2c+3$ for some $c \in [1,2]$. First note $f(1)=5$ and $f(2)=7$. By the IVT, all values $c \in [1,2]$ are hit. I'm just wondering how to put all of these facts together to arrive at…
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