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
4
votes
2 answers

Limited function.

Let $f:[0,+\infty) \rightarrow \mathbb{R}$ be a function bounded on each bounded interval. Prove that if $\lim_{x \rightarrow +\infty } [f(x+1)-f(x)] = L$, then $\lim_{x \rightarrow +\infty} \frac{f(x)}{x} = L$.
Croos
  • 1,809
4
votes
2 answers

Set of all functions with a Lipschitz Condition

I could not find a way to start with, let alone solution. Any help would be greatly appreciated. Let $M_K$ be the set of all functions f in $C_{[a,b]}$ satisfying a Lipschitz condition i.e., $|f(t_1)-f(t_2)|\leq K|t_1-t_2|$ for all…
user64066
  • 2,480
4
votes
2 answers

if a set is a countable union of compact sets, is it closed?

I know that if a set is closed then it can be expressed as a countable union of compact sets. But is the converse true? I am asking because while reading a real analysis book, the following comment is remarked: Let $\bar{B_k}$ the the closure of a…
Tomas Jorovic
  • 3,983
  • 3
  • 27
  • 38
4
votes
3 answers

$x_{n+1}\le x_n+\frac{1}{n^2}$ for all $n\ge 1$ then did $(x_n)$ converges?

If $(x_n)$ is a sequence of nonnegative real numbers such that $x_{n+1}\le x_n+\frac{1}{n^2}$ for all $n\ge 1$ then did $(x_n)$ converges? Can someone help me please?
sumon
  • 49
4
votes
1 answer

diffeomorphism between $F^{-1}(0,1)$ and $S^2$ sphere

I have problem with this: Prove that $F^{-1}(0,1)$ and $S^2$ are diffeomorphic, where $$F(x,y,s,t)=(x^2+y,x^2+y^2+s^2+t^2+y).$$ I have found that $$F^{-1}(\{0,1\})=\{ (x,y,s,t) \in \mathbb{R}^4: y^2+s^2+t^2=1, \ x^2+y=0 \}$$ but I can't find…
amoneth
  • 51
4
votes
3 answers

Clear explanation about uniform continuity.

Can anyone explain the uniform continuity clearly with picture if possible?? I have read the section on this topic in my text book but I am still not clear on this. Thanks. --edit In my text book, it gives two definitions of a continuous function.…
eChung00
  • 2,963
  • 8
  • 29
  • 42
4
votes
3 answers

Int(E) mean ? what does Int mean

In real analysis what is Int symbol mean? like Int(E) , int (A U B) ?? I want to know what is Int mean in real analysis a few example will be also good thank you
james Miler
  • 349
4
votes
1 answer

Real Analysis question in Second Fundamental Theory of Calc

Define $$F(x)=\int_1^x \frac{1}{2\sqrt{t}-1} dt \quad \text{for all $x\ge 1$}.$$ Prove that if c>0, then there is a unique solution to the equation $$F(x)=c, \quad x>1.$$ Attempt at a solution: I am not sure how to "prove" this. What I have so far…
Marie
  • 103
4
votes
4 answers

Construct a compact set of real numbers whose limit points form a countably infinite set.

I have seen examples of sets that have these properties, like: $$A=\left\{\frac1n+\frac1 m:m,n\in\Bbb N\right\}\cup\{0\}$$ And it is clear that 0 and all 1/n are limit points. However, how does one show that there are no other limit points? I am…
mb7744
  • 1,292
4
votes
1 answer

Convex Function Inequality

Let $f: I \to \mathbb{R}$ where $I$ is an interval. We say that $f$ is convex if for every $a,b\in I$ and every $\lambda : 0<\lambda < 1.\\$ Prove that for any $x\in (a,b)$ $f(\lambda b + (1-\lambda)a) \leq \lambda f(b) + (1-\lambda )f(a) \implies…
Kyle H.
  • 293
  • 1
  • 3
  • 7
4
votes
1 answer

What are good examples for functions that are semi-continuous but not left/right continuous and vice versa?

I am looking for good simple examples that capture the difference in definitions between semi-continuity and continuity from the left/right. I am certainly able to construct examples and wikipedia also lists…
Phira
  • 20,860
4
votes
1 answer

Image of measurable set under continuous inverse function is always measurable?

Let $f$ be a continuous function on a set $E$. Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable? Is this correct?
Yang
  • 521
  • 1
  • 9
  • 18
4
votes
1 answer

How to prove $\| \int_X f \| \leq \int_X \| f \|$ in higher dimension?

Let $E$ be a finite dimensional real vector space with a norm $\|.\|$. Define integral of mesurable functions with value in $E$ by choosing a basis and integrate componentwise. How do we prove the triangle inequality : $$ \left\| \int_X f \right\|…
user10676
  • 8,521
4
votes
2 answers

a basic doubt on the theorem that $f$ is continuous iff the inverse image of every open set is open

Suppose $f:X \to Y$ and some "not open set" in $X$ is the inverse image of an open set in $Y$. then the function is not continuous as there is an open set whose inverse image is not open. But, intuitively thinking, what is the reason behind this ?…
user96000
  • 353
4
votes
1 answer

Rough bounds on the value

In one book the author defines the following value: $$ F(m) = \sup\limits_{v\geq 1}\frac{v^m}{e^v}\int\limits_1^v\frac{e^u}{u^m}\mathrm du $$ for $m\in [3,4]$. Further he puts an upper bound for this value: $$ F(m)\leq 1+\frac m2+\frac…
SBF
  • 36,041
1 2 3
…
99 100