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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Differential Calculus - Swapping a sup and a limit.

Well, I'm doing some exercises on differential calculus and I'm stuck. (a) Let $U \subset \mathbb R^m$ and $f: U \rightarrow \mathbb R^n$ be a continuous function on a line segment $[x, x+h]\subset U$ and differentiable on $]x, x+h[$. Show that if…
user79594
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Lower Bound on Log function

In this paper "Papandriopoulos, J.; Evans, J.S., "SCALE: A Low-Complexity Distributed Protocol for Spectrum Balancing in Multiuser DSL Networks," Information Theory, IEEE Transactions on , vol.55, no.8, pp.3711,3724, Aug. 2009" The authors used the…
Amr
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Lipschitz function and uniform continuity on the real line

Let $\phi\colon\mathbb{R}\to\mathbb{R}$ be a function that satisfies $\lvert \phi(x)\rvert \geq 1$ for all real $x$ and $\lvert\phi(x) - \phi(y)\rvert \leq \lvert x - y\rvert $ for all real $x,y$ (for example $\phi(x) = \cos(x)$). Let $f(x) =…
Anon
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Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0

Let f : [0, 1] → R be a continuous function with f (0) = 0 and suppose that it is differentiable for all x ∈ (0, 1). Further assume that $x \mapsto f'(x)$ is a strictly increasing function on (0, 1). Show that $x \mapsto \frac{f(x)}{x}$ is also a…
Anon
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Limit of the derivative of a nonnegative decreasing function

Let $f:[0,\infty) \to [0,\infty)$ be a $C^2$ and monotone decreasing function. I think it can be shown that this condition implies that $\limsup_{x\to\infty} f'(x) = 0$. However, I don't think that the same is in general true for the $\liminf$. But…
user1736
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Open and Closed Sets Examples

Ok so well Im struggling to find examples for the first two parts and for the last, well I don't think it is open but can't again find an example. Thanks.
WhizKid
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Is it possible to generalize the Mean value theorem for integral not on compact set?

I wonder it is possible to extend the mean value theorem not on compactness. In more detail, Let $f : A \rightarrow \mathbb{R}$ be continuous on $A \subset \mathbb{R}^n$. The mean value theorem for integral states that if $A$ is connected and…
Block Jeong
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Why a uniform limit of a sequence of bounded derivative is a derivative?

Assume that a sequence $(f_n)$ of functions $f_n:[0,1] \rightarrow \mathbb R$ is uniformly convergent to a function $f:[0,1]\rightarrow \mathbb R$. Moreover let (for each $n\in \mathbb N$) $f_n$ be bounded and $f_n$ be a derivative of some $F_n:…
A.B
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How prove this the equation $f'(x)+f(x)=0$ has at least one root in $(a,b)$

Suppose $f$ is continous on $[a,b]$ and differentiable on the open interval $(a,b)$,and such $$f(a)+f(b)=0$$ show that: then the equation $f'(x)+f(x)=0$ has at least one root in $(a,b)$ My try: not $$(e^xf(x))'=e^x(f(x)+f'(x))$$ But I can't,Thank…
user94270
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Prove that $\lim_{n \to \infty} \int_0^2 e^{ x^2 / n}\,{\rm d}x$ exists and evaluate it.

I need to show that this limit exists and then evaluate it. It is from a section on uniform convergence of sequences. I know that if $f_n \rightarrow f$ uniformly and each $f_n$ is integrable, then I can bring the limit inside of the integral. I'm…
MFD55012
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Is $f(n) = 2^{\frac{1}{2}(n^2-n)} / n!$ polynomially bounded?

The numerator counts the number of different adjacency matrices. I think Sterlings approximation helps to anwser my question but I fail to derive the answer. So, is there a polynomial function $g(x)$ such that $$f(x) \leq g(x)$$
John D.
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if $2f(x)+f''(x)=-xf'(x)$ show that $f(x)$and $f'(x)$ are bounded on $R$

Assume that $f$ is twice diffentiable on $R$,and such $$2f(x)+f''(x)=-xf'(x)$$ show that: $f(x)$and $f'(x)$ are bounded on $R$ My try:since $$2f(x)+f''(x)+xf'(x)=0$$ and following I can't any work,Thank you very much!
user94270
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How prove $f'(x)$ must bounded

let $f(x)$ have two derivatives,and $c$ is constant, for all real $x$ such $$f(x)(x^2-1)+xf'(x)=c$$ show that:$f'(x)$ is bounded function,In other words: there exist $M>0$,such $|f'(x)|\le M,x\in R$ My try:if we take the…
math110
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Let $f(x)=|x|$ for $x$ rational and $f(x)=1$ for $x$ irrational. Show $f$ has limits at $1$ and $-1$. Find them.

Let $f(x)=|x|$ for $x\in\mathbb{Q}$ and $f(x)=1$ for $x\in\mathbb{R}/\mathbb{Q}$, where $\mathbb{Q}$ is the set of rationals. (a) Show $f$ has limits at $1$ and $-1$. Find them. (b) Show that if $c\not=1$ or $c\not=-1$, then $f$ doesn't have a…
Desperate Fluffy
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Totally bounded metric space

Consider $\Bbb{R}$ with the following metric: $$\rho(x,y):=\left|\frac{x}{1+|x|}-\frac{y}{1+|y |}\right|$$ Then I wish to show that $(\Bbb{R}, \rho)$ is a totally bounded metric space. Of course by totally bounded I mean that …
George
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