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
2 answers

Lebesgue integral, open set and closure

This is one of the problem in my final exam. The following integral is talked in the sense of Lebesgue's. Suppose $f$ is integrable on the real line. If for any open set $G\subset R$, $$\int_Gfdx=\int_{\overline{G}}fdx$$ Then $f = 0,a.e.$ I managed…
R. Feng
  • 819
5
votes
4 answers

How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?

How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?
user7485
5
votes
3 answers

Problem with a limit and the infinite intersection of open sets

This is probably a really stupid question, but it's been annoying me for a while and I still can't find an answer that convinces me. We know that $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$. But, in my lecture, we saw that $\{0\} =…
justdoit
  • 677
  • 1
  • 7
  • 17
5
votes
3 answers

The function $F(y)=\int_0^\infty e^{-x^2}\cos(2xy)dx$ (Lebesgue) satisfies $F'(y)+2yF(y)=0.$

I want to prove that the function defined as the Lebesgue Integral $$F(y)=\int_0^\infty e^{-x^2}\cos(2xy)dx$$ satisfies $F'(y)+2yF(y)=0$, and after that, that $F(y)=\frac{1}{2}\sqrt{\pi}e^{-y^2}$. I tried this: First, we have that…
LeviathanTheEsper
  • 3,163
5
votes
3 answers

Show $f$ has exactly one fixed point if $f'(x) <1$ for all $x$

I'm trying to use the mean value theorem to show that for differentiable $f:\mathbb{R} \to \mathbb{R}$, if $f'(x) < 1 \forall x\in\mathbb{R}$, then $f$ has exactly one fixed point. I know how to show that if $f'(x)\neq 1$ then $f$ can have at most…
TonicWaterWithLime
  • 73
5
votes
1 answer

The philosophy of change of variables

Change of variables is a basic method in mathematical solving. However, I can't use it smoothly, i.e. I don't know when to use it. (I can use it in some regular problems, but I won't have the sense of using it when coming across some new problems).…
89085731
  • 7,614
5
votes
1 answer

Am I missing something quite obvious from understanding this proof?

Theorem. There exists a nonempty set of rational numbers which is bounded above in $\mathbb{Q}$ but has no least upper bound in $\mathbb{Q}$. Here is the relevant part of the proof that I am going to ask about. Let $k=\frac{a}{b}\in\mathbb{Q}$ be…
user265696
5
votes
4 answers

Prove that $f$, such that $\forall\delta\gt0$ $|f(y)-f(x)|\lt\delta^2$ $\forall x,y\in\mathbb{R}$ and $|y-x|\lt\delta$, is a constant function.

The function $f$ is defined on $\mathbb{R}$ such that for every $\delta\gt0$, $|f(y)-f(x)|\lt\delta^2$ for all $x,y\in\mathbb{R}$ and $|y-x|\lt\delta$. Prove that $f$ is a constant function. So, what I know, is that I need to show that $f(a)=f(b)$…
Jeroen
  • 1,653
5
votes
3 answers

Mean Value Theorem for Multiple Variables

Is there any reason that this generalization of the mean value theorem would fail? Let A be a subset of Rn that is differentiably connected, and let f : A --> R be continuously differentiable at every point in A. If x and y are two points in A then…
user7060
5
votes
5 answers

show that $\lim_{x\rightarrow 0} \left(\frac{\sin(x)}{x}\right)^{1/x^2} = e^{-1/6}$

I see that this is indeterminate form, so I approach L'Hopital rule, but I can not find this limit. Please help me to find this limit.
Saroj Ghosh
  • 353
5
votes
4 answers

How to prove proposition $2.2.14$ in Tao Analysis $I$?

Proposition $2.2.14$ (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m \geq m_0$, we have the following implication: if $P(m')$ is…
Student
  • 9,196
  • 8
  • 35
  • 81
5
votes
1 answer

Prove that if $x \leq \vert 2 \vert$ then $ \vert x^2-4 \vert \leq 4 \vert x-2 \vert$

This is a problem on my first homework from Real Analysis Assume that $\vert x \vert \leq 2$ Proof: $\vert x \vert + \vert2\vert \leq 2 + \vert 2 \vert \to \vert x+2 \vert \leq 4$, add 2 to each side $\vert x-2 \vert \vert x+2 \vert \leq 4 \vert x-2…
K. Gibson
  • 2,381
5
votes
2 answers

A strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ is continuous at least one point.

Let $f$ be a strictly increasing function (that is, $f(b) > f(a)$ if $b > a$). Show that $f$ is continuous at at some point. Hint: Use the fact that uncountably many positive real numbers can not have a finite sum. This leads me to the following…
Nitin
  • 2,948
5
votes
2 answers

Tricks to Intuitively See Uniform Continuity

If one gives you a continuous function $f(x)$ on a set, let's say the interval $[a, b]$. Are there any fast ways to intuitively see if $f(x)$ is uniformly continuous on $[a, b]$?
user198
  • 71
5
votes
3 answers

Minimum of $f(x)=\sqrt{9x^{2}+1}-3x-2 $

Let $$f(x)=\sqrt{9x^{2}+1}-3x-2 $$ Show that $f$ is bounded from below by $-2$ $$\forall x\in \mathbb{R}\quad f(x)>-2 $$ Is $-2$ the minimum value of $f$ ? Indeed, let $x\in \mathbb{R}$ \begin{align} f(x)>-2 &\iff \sqrt{9x^{2}+1}-3x-2>-2\\ &\iff…
Yacob
  • 447
1 2 3
…
99 100