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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What does 'finite-valued' mean?

In Rudin, while defining the concept of 'pointwise bounded', it says: if there exists a finite-valuded function $\phi$ defined on $E$ such that $|f_{n}(x)|<\phi (x)$. Here, I am quite puzzled by the definition of finite-valued. Is $f(x)=1/x$ an…
Martin
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A problem related to the submarine puzzle

Submarine puzzle: A submarine is located at an integer somewhere along the number line. It is moving at some integral velocity (an integral number of units per second). Every second you may drop a bomb which will destroy the submarine if the…
nnkken
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about periodic functions

Could anyone give some idea about the following problem? Many thanks！ Suppose that $f,g: \mathbb{R}\to\mathbb{R}$ are two periodic functions such that $\lim_{x\to\infty}[f(x)-g(x)]=0$. Show that $f(x)=g(x)$ for all $x\in\mathbb{R}$.
OnoL
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Proving this function $f:[0,1]\to \mathbb{R}$ is continuous on $[0,1]$

I am looking for a hint or feedback on what I've already done, not a full solution. $f=t\sin{\left(\frac{1}{t}\right)}$ for $t\ne 0$, $f(0)=0$, My idea is that I only have to worry about the steep parts. My approach: Proof sketch: Let $\epsilon > 0…
Rustyn
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Functional equation $f\left( \dfrac{2xy}{x+y}\right) +f\left( \dfrac{x+y}{2}\right) =f\left( x\right) +f\left( y\right)$

I have not any idea, how to attack the equation $$f\left( \dfrac{2xy}{x+y}\right) +f\left( \dfrac{x+y}{2}\right) =f\left( x\right) +f\left( y\right)$$ with unknown $f:\mathbb{R} _{+}\rightarrow \mathbb{R}$. Allowing (for a while) that $y$ can be…
user472228
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$f''(x)+e^xf(x)=0$ , prove $f(x)$ is bounded

Differentiable function in $\mathbb{R}$ for which $f''(x) + e^x f(x)=0$ for every $x$. Prove that $f(x)$ is bounded as $x \rightarrow +\infty$ I have tried a few stuff but they didnt work out, for example i noticed that the function has infinite max…
Plom
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Show that there is no continuously differentiable function on closed disks

In Cartesian coordinates, the closed unit disk is given by $$ \bar{D}=\left\{(x, y) \mid x^2+y^2 \leq 1\right\}, $$ and its boundary is given by $$ \partial D=\left\{(x, y) \mid x^2+y^2=1\right\}. $$ Prove that there is no continuously…
unicornki
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Homework: Smooth mapping $f$ satisfying $f\circ f=f$

This is an exercise in Mathematical Analysis by Zorich, in the subsection 12.1. Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a smooth mapping satisfying condition $f\circ f=f$. $\quad$a) Show that the set $f(\mathbb{R}^n)$ is a smooth surface in…
Wei Zhan
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Are there any function which's derivate is the scaled one of the original?( $f'(x) = f(cx)$ )

which function could satisfy the following, for a certain $c\ne1$ $f'(x) = f(cx)$ ...beyond the trivial $f=0$ i've been thinking about it for a while. for a simpler case: $f'(x)= f(x+c)$ i've found $e^{xe^v}$ where $v$ is the solution of…
Zoltán Haindrich
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Rudin Theorem 2.41 - Heine-Borel Theorem

When proving Theorem 2.41 in Principles of Mathematical Analysis: Let $E \subset \mathbb{R}^k$. If every infinite subset of $E$ has a limit point in $E$, then $E$ is closed and bounded. Rudin says, "If $E$ is not bounded then $E$ contains points…
user70962
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Can we deduce strictly increasing on a sub-interval for such a function?

Assume that we have a function $f : [0, 1] \rightarrow [0, 1]$ which is differentiable, $f(0) = 0$ and $f(1)=1$. Can we deduce that there exists an open sub-interval in $[0, 1]$ in which $f$ is strictly increasing over that? Note that here,…
Mohammad Ali Nematollahi
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Proof to show function f satisfies Lipschitz condition when derivatives f' exist and are continuous

The question is as follows: Given a function $f$, 2 known information: (1) $f'(x)$ exist (2) $f'(x)$ are continuous Goal: function $f$ satisfies Lipschitz condition on any bounded interval $[a,b]$ Here is my attempt: 1/ Recall…
Cecile
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Why isn't Volterra's function Riemann integrable?

My construction of Volterra's function is as follows. Let $F(x)=\begin{cases} x^2\sin\left(\frac{1}{x}\right) &\text{ if } x \neq 0\\ 0 &\text { if } x =0 \end{cases}$ On the interval $\left[0,\frac{1}{8}\right]$ we find the most extreme value of…
emka
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Rectifiable functions

A continuous function $\alpha: [a,b] \to \mathbb{R}^k$ is called a curve. For each partition $P = \{t_0
mary
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Application of Weierstrass theorem

Let f be a continuously differentiable function on $[a.b]$. Show that there is a sequence of polynomials $\{P_n\}$ such that $P_n(x) \to f$ and $P'_n(x) \to f' (x)$ uniformly on $[a,b]$ My approach has been as follows. Since f is continuously…
sarahz
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