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
1 answer

Can a non-constant two dimensional polynomial have a set of zero points of positive measure?

Can a non-constant two dimensional polynomial $P(x,y): \mathbb{R}^2\rightarrow \mathbb{R}$ have a set of zero points of positive measure? In complex analysis I know that a non-constant analytic function can't have an isolated point. Here can we…
Shine
  • 3,025
  • 13
  • 26
4
votes
2 answers

Sufficient Condition for $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$

Suppose that $a_n$ and $b_n$ have a finite limit. Then is it true that $\lim_{n \rightarrow\infty} \frac{b_n}{a_n}=1$ is enough to ensure that $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$? Attempt: If $\lim_{n \rightarrow\infty} \frac{b_n}{a_n}=1$…
recmath
  • 2,738
4
votes
4 answers

prove that $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}|f'(x)|dx$

I would appreciate if somebody could help me with the following problem Q: prove that $$\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}|f'(x)|dx$$ where $f'(x)$ is continuous on $(0,1)$, and $f(0)=0$.
Young
  • 5,492
4
votes
1 answer

Determining the relevant set/field for least upper bounds

My real analysis textbook makes the following statement: a set of real numbers $S_1 = \{ s \in \mathbb{R} | s^2 < 2 \}$ has a least upper bound, as we can show that $sup S_1 = \sqrt{2}.$ On the other hand, it contends that $S_2 = \{ s \in \mathbb{Q}…
reallllyconfused
  • 43
4
votes
1 answer

How to show that $H(x)=\int_0^1 f(y-x)g(y)~dy$ is bounded and continuous on $\mathbf{R}$

Given that $f$ and $g$ belong to $L^2(\mathbf{R})$, how can I show that $$ H(x)=\int_0^1 f(y-x)g(y)~dy$$ is a bounded and continuous function on $\mathbf{R}$. My attempt for the boundedness part: $$\begin{align*} |H(x)| = \left|\int_0^1 …
Jack
  • 763
4
votes
2 answers

Can the concept of a jump discontinuity be extended to functions of the form $f\colon \mathbb{R}^2 \to \mathbb{R}$

We know that for a function $f \colon \mathbb{R} \to \mathbb{R}$, a jump discontinuity at a point $P$ is defined as the left and right limits exist but not equal. I'd like to know if this concept can be extended to functions of the form…
Rajesh D
  • 4,247
4
votes
3 answers

An application of Weierstrass theorem?

I'm going through some problems and I'm really stumped on this one. The questions says that Given $f(x)=|x|$, show that there is a sequence of (real) polynomials $P_n(x)$ with $P_n(0)=0$ that converge uniformly to $f(x)$ on the interval…
Cindy
  • 97
4
votes
3 answers

For $f, g \in C^1$, $fg' - f'g \neq 0$ implies that the zeros interlace

Let $f, g \in C^1$, and suppose that $f(x) g'(x) - f'(x) g(x) \neq 0$ for all $x$. Show that The roots of $f$ do not have an accumulation point. The roots of $f$ and $g$ interlace, so that if $f(x_0) = f(x_1) = 0$ with $f(x) \neq 0$ for $x \in…
user182973
  • 475
  • 2
  • 8
4
votes
1 answer

Proving this function is differentiable at $1$

Define $h(x) = 1$ except at $1$ where $h(1) = 0$. Also define $H(x) = \int_0^x h(t)$. Now I tried to show that $H$ is differentiable at $1$. My proof is to compute $$ \lim_{x \to 1^-} {H(1) - H(x)\over 1-x} = 1$$ and $$ \lim_{x \to 1^+} {H(x) -…
blue
  • 2,884
4
votes
1 answer

Every uncountable subset of $\mathbb{R}$ has a limit point

I am looking at this problem and I decided to attack it by proving the contrapositive. If $E \subset \mathbb{R}$ has finitely limit points, then $E$ is countable. My proof: Let $S=\{\text{limit points of E} \}$, then for any $s\in E, \ \exists…
Mr.Fry
  • 5,003
  • 3
  • 19
  • 28
4
votes
1 answer

how to prove $\int_\mathbb{R} |f(x+h)-f(x)|dx\leq A|h|$?

Let $f$ be of bounded variation on $\mathbb{R}$, i.e it is of bounded variation on any finite subinterval $[a,b]$. define $$A=\sup_{a, \ \ b}V_a^b(f)<\infty$$Here $V_a^b(f)$ denotes the total variation of $f$ over the interval $[a,b]$. How to show…
Shine
  • 3,025
  • 13
  • 26
4
votes
1 answer

Show that Laplace transform is differentiable

I have the following homework problem that is quite confusing. Let $I$ be an open subinterval of $\mathbb{R}$, and let $f:\mathbb{R} \to \mathbb{R}$ be a Borel measurable function such that $x \mapsto \exp^{tx}\,f(x)$ is Lebesgue integrable for…
nate
  • 2,159
4
votes
1 answer

Operator such as $-1$ is the identity element

Short question Do you know an operator such as $-1$ is the identity element ? Long Question This morning, I had a hard time with identity elements. I'm pretty sure that the following isn't very rigourous, so please don't hesitate to comment ! I'm…
Fabien
  • 2,958
4
votes
0 answers

Arclength integral

Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative $f'$ whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for every $n \in \mathrm N$, there exists a tagged…
Daniel
  • 115
4
votes
1 answer

Convergence in $L^\infty$ is nearly uniform convergence

Let $\{f_n\}$ be a sequence of functions in $L^\infty$. I want to prove that $\{f_n\}$ converges to $f\in L^\infty$ $\Leftrightarrow$ there is a set $E$ of measure zero such that $f_n$ converges uniformly to $f$ on $E^c$. My…
Colin
  • 703
1 2 3
…
99 100