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

Suppose that $x$ is a fixed nonnegative real number such that for all positive real numbers $E$ , $0\leq x\leq E$. Show that $x=0$.

Suppose that x is a fixed nonnegative real number such that for all positive real numbers $E$ , $0\leq x\leq E$. Show that $x=0$.
Man
  • 169
5
votes
2 answers

If $f$ has antiderivative, must $|f|$ also have antiderivative?

Suppose function $f:[a,b] \to \mathbb{R}$ has an antiderivative. This means there is some differentiable function $F$ so that $f(x) = F'(x)$ in all of $[a,b]$. Must the absolute value $|f|$ have an antiderivative? What if we also specify that $f$…
SAS
  • 1,066
  • 9
  • 18
5
votes
0 answers

$g(x)$ be a continuous function with recursive sequence

Let $g:[0,\frac{1}{2}]\to \mathbb{R}$ be a continuous function.Define $g_n:[0,\frac{1}{2}] \to \mathbb{R}$ by $g_1=g $ and $g_{n+1}(t)=\int_0^t g_n(s)ds\, \forall n\ge 1$.Show that $\lim_{n\to \infty}n!g_n(t)=0 \, \forall t\in [0,\frac{1}{2}]$ i…
RAM_3R
  • 1,256
5
votes
2 answers

Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$?

Does the series $\sum \frac{x^n}{1+x^n}$ converge uniformly on $x\in [0,1)$? I have no idea where to start. Could somebody give me a hint ? Edit: Could I use something like this ? $$\left|\frac{x^n}{1+x^n}\right|\leq x^n$$ Because $x<1$, the…
Kasper
  • 13,528
5
votes
4 answers

Suppose that $x > 1$. Prove that $s_n \to 1$

Question: Suppose $x > 1$. Prove that $x^\frac{1}{n} \to 1$. The following is a list of what I am trying to do through out my proof. Show $s_n$ is monotone (decreasing). Show $s_n$ is bounded. Using the monotone convergence thm and thm 19, show…
enlgmatlc
  • 479
5
votes
5 answers

Let $x_n=\sqrt[3]{n+1}-\sqrt[3]{n}$. Prove that $(x_n)$ converges.

Let $x_n=\sqrt[3]{n+1}-\sqrt[3]{n}$. Prove that $(x_n)$ converges. I managed to prove the sequence converges by using definition of convergence, but initially I thought of using the Monotonic Convergence Theorem to prove it. However, I'm stuck at…
Idonknow
  • 15,643
5
votes
3 answers

Prove that $f(x)=\arcsin(x)+\arccos(x)$ is constant function and it is $f(x)= \frac{\pi}{2}$

I know that the best way is using infinitesimal calculus but this way we use on the lecture and then we must prove it in the other way which don't use infinitesimal calculus. My try:$\lim_{ x\to 1^{-} }\arcsin(x))+\arccos(x))=\lim_{ x\to -1^{+}…
MP3129
  • 3,195
5
votes
2 answers

Discontinuous function with continuous inverse?

Here is something that is confusing me. The function $\begin{equation} f(x)=\begin{cases} x, & \text{if $x \in[-1,0]$}\\ x+1, & \text{if $x \in (0,1]$} \end{cases} \end{equation}$ Is clearly discontinuous at $x=0$ yet its…
Noble.
  • 2,446
5
votes
2 answers

When does a function take an open interval into an open interval?

Suppose we consider a function $f(x)$ defined from an open interval say $(a,b)$ to some set $T$. When would the set $T$ be an open interval?
Manjil P. Saikia
  • 562
5
votes
4 answers

Is this property true?

Suppose $f$ is a continuous function on the compact set $[0,1]$. Is true that given $\epsilon>0$, there exist $\delta>0$ such that for any partition $P=\{x=x_0
Tomás
  • 22,559
5
votes
3 answers

Show that the function $f(x)=e^{-\frac{1}{x}},\text{ if }x>0\wedge f(x)=0,\text{ if }x\leq 0$ is smooth and all derivatives vanish in $x=0$

I had an idea but I think it is wrong. I have said $e^z$ is smooth because the n-th derivative of $e^z$ is always $e^z$ and then I have substituted $z$ with $-\frac{1}{x}$. But just calculating the first derivative Shows me that I am probably wrong…
New2Math
  • 1,265
5
votes
3 answers

Understanding why the author chose the number he did in this proof that $\sqrt 2$ exists

I am reading a proof for the existence of $\sqrt 2$. The first half of the proof goes as follows: Consider the set $T = \{t \in \mathbb{R} : t^2 \lt 2\}$. Let $\alpha = \sup T$. Case 1: Show $\alpha^2 \lt 2$ is impossible by implying $\alpha$ is…
CL40
  • 411
5
votes
1 answer

Let $a_n=\frac {1}{3^n}$ if $n$ is prime and $a_n=\frac {1}{4^n}$ if $n$ is not prime.

I am stuck on the following problem that says: Let $a_n=\frac {1}{3^n}$ if $n$ is prime and $a_n=\frac {1}{4^n}$ if $n$ is not prime. Then I have to find the radius of convergence of the power series $$\sum_{n=1}^{\infty} a_nx^n$$ ? I know that…
user52976
5
votes
2 answers

Total variation of a step function

What is the formula for the total variation of a step function on [a,b]? I understand how to write a formula for the total variation of a general function of bounded variation. Any ideas?
Libertron
  • 4,415
5
votes
2 answers

proof continuity of a multivariable function

$$ f(x,y) = \begin{cases} \frac{x^3-xy^2}{x^2+y^2}, & \text{if }(x,y)\not= (0,0) \\ 0, & \text{if } (x,y)=(0,0) \end{cases} $$ how to prove $f(x,y)$ is continuous on $\mathbb{R}^2$? I know it must be continuous on $\mathbb{R}^2$, but according to…
i_a_n
  • 755
1 2 3
…
99 100