Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
5
votes
3 answers

simplifying an expression which concludes $e^x$

I am solving an high school question right now(well, I'm an high-scholar) and I'm don't understand how they simplify the expression: $$\frac{-e^{x}+\frac{e^{2x}}{\sqrt{1+e^{2x}}}}{-e^{x}+\sqrt{1+e^{2x}}}$$ To be the expression…
Neil Rauch
  • 129
5
votes
1 answer

If $\int_{\mathbb{R}}\vert f\vert<\infty$, then $\sum_{n=1}^\infty\vert n\int_n^{n+1/n}f(x+y)dy\vert<\infty$

We are given $\int_{\mathbb{R}}\vert f\vert <\infty$. Want to show that for almost every $x$, $\sum_{n=1}^\infty\vert n\int_n^{n+1/n}f(x+y)dy\vert<\infty$. I have two ideas. The first is to let $f_n=\int_n^{n+1/n}\vert f\vert$. I can show by DCT…
cap
  • 2,443
5
votes
3 answers

Sequence of Rationals Converging to a Limit

I'm trying to show that for every real number $r$, there exists a sequence of rational numbers $\{q_{n}\}$ such that $q_{n} \rightarrow r$. Could I get some comments on my proof? I know that between 2 reals $r, b$ there exists a rational number $m$…
user26139
  • 279
5
votes
1 answer

Prove that there exists a monotone nondecreasing function $f:[0,1] \rightarrow \mathbb{R}$ discontinuous in rationals

Prove that there exists a monotone nondecreasing function $f:[0,1] \rightarrow \mathbb{R}$ discontinuous in rationals of $[0,1]$ I tried to test functions of the type $$ f(x)=\begin{cases} \frac{1}{q} & \text{if } x=\frac{p}{q} \mid p,q \in…
sebastian azocar
  • 293
5
votes
2 answers

Can a number be normal in an arbitrary set of bases?

Suppose we have a subset $S$ of the natural numbers greater than or equal to $2$. Is there a real number $x$ such that $x$ is $k$-normal for all $k$ in $S$, and not $k$-normal for all $k$ in the complement of $S$, no matter what $S$ we choose?
user107952
  • 20,508
5
votes
1 answer

Differentiable functions dense in Lipschitz functions?

Consider $C^1([0,1])$ the functions with continuous derivative on $[0,1]$ (one-sided derivatives at each end), and $\operatorname{Lip}([0,1])$ the Lipschitz functions on $[0,1]$. The mean value theorem of course shows that $C^1([0,1]) \subseteq…
Matthew Daws
  • 2,825
5
votes
0 answers

What is uniform integrability?

I know two definitions of uniform integrability: (Rudin - Real and Complex Analysis). Let $(X,\mathfrak{M},\mu)$ be a positive measure space. A set $\Phi\subset L^1(\mu)$ is called uniformly integrable if to each $\epsilon>0$ corresponds a…
Peter
  • 51
5
votes
5 answers

If $f$ is differentiable on $[a,b]$, then it is also Lipschitz on it.

He guys, I am trying to show that a differentiable function defined on a closed interval is also Lipschitz on it. I managed to weave the below proof, but I have a feeling that it may be just a tad too general for this purpose: Theorem. If $f$ is…
wjmolina
  • 6,218
  • 5
  • 45
  • 96
5
votes
3 answers

Differentiable function for which the tangent at each point has infinitely many common points with the graph

There exists a differentiable function $f\colon\mathbb{R}\to\mathbb{R}$ with the following property: the tangent at each point has infinitely many common points with the graph /Edit: $f$ nonlinear/ For $\sin x^2$ we have this property at point…
larry01
  • 1,816
5
votes
1 answer

Given $f: I \rightarrow \mathbb{R} $ differentiable and $c \in I$, are there $a,b \in I$ such that $f(b) - f(a) = f '(c) (b - a)$?

Given $f: I \rightarrow \mathbb{R} $ differentiable (where $I$ is an interval) and $c \in I$, are there $a,b \in I$ such that $f(b) - f(a) = f '(c) (b - a)$? Sort of like the reverse of the Mean Value Theorem. I'm not sure if it's true, but I feel…
violeta
  • 797
5
votes
0 answers

Every $f\in L^\infty(\mu)$ is a uniform limit of simple functions $f_i$.

Problem: Let $\mu$ be a positive (finite?) measure on a space $X$. Show that every $f\in L^\infty(\mu)$ is a uniform limit of simple functions $f_i$. This question came while reading a proof in Rudin's Real and Complex Analysis (Theorem 6.16). It…
Cachalot
  • 51
5
votes
1 answer

When is a real function orthogonal to its derivative?

I saw this proof that a function $f$ is orthogonal to its derivative $f'$: $$ \int_{-\infty}^\infty f(t)f'(t)dt = \frac{1}{2\pi} \int_{-\infty}^\infty F(\Omega) (-j\Omega) F^*(\Omega) d\Omega = -\frac{1}{2\pi} \int_{-\infty}^\infty j\Omega…
Mark
  • 195
  • 1
  • 5
5
votes
3 answers

Applying the Stone-Weierstrass Theorem to approximate even functions

Let $f:[-1,1] \rightarrow \mathbb{R}$ be any even continuous function on $[-1,1]$ (i.e. $f(-x)=f(x)$ $\forall x \in [-1,1]$). Let $\epsilon>0$. Prove that there exists an even polynomial $p$ such that $$|f(x)-p(x)|< \epsilon$$ $$\forall x \in…
Ducky
  • 2,006
5
votes
2 answers

A question regarding a problem in Folland

Let $f_n(x) = ae^{-nax}-be^{-nbx}$ where $0
Keith
  • 7,673
5
votes
2 answers

Is complement of a dense set in $\mathbb{R}$ dense in $\mathbb{R}$?

$\mathbb{Q}$ is dense in $\mathbb{R}$. Also, its complement, $\mathbb{R-Q}$, is dense in $\mathbb{R}$. I know that we can proof denseness of $\mathbb{Q}$ and $\mathbb{R-Q}$ separately for each of them. Is it true that complement of EVERY dense set…
user200918
1 2 3
…
99 100