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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. .

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Lipschitz function maps set of measure zero to set of measure zero

Let the function $f: [a, b] \rightarrow \mathbb{R}$ be Lipschitz, that is, there is a constant $c \geq 0$ such that for all $u, v \in [a, b]$, $|f( u) - f( v)| \le c |u - v|$. Show that $f$ maps a set of measure zero onto a set of measure zero. The…
Idonknow
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Show that $f$ has at most one fixed point

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a differentiable function. $x\in\mathbb{R}$ is a fixed point of $f$ if $f(x)=x$. Show that if $f'(t)\neq 1\;\forall\;t\in\mathbb{R}$, then $f$ has at most one fixed point. My biggest problem with this is that…
chris
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Closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed

I am looking for closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed. I define $A+B=\{a+b:a\in A,b\in B\}$ I thought of this example, but it is only in $\mathbb{R}$. Take: $A=\{\frac{1}{n}:n\in\mathbb{Z^+}\}\cup\{0\}$ and…
Moss
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Showing a function of two variables is measurable

Let f(x,y) be a function defined on the unit square $0\leq x\leq1$, $0\leq y\leq1$ that is continuous on each variable separately. Is f a measurable function of (x,y)? I think I need to look at the pre-images of f, and I need to use the fact that…
Ashley
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How do you define sequences that converge to infinity?

For instance consider the sequence $\{1,0,2,0,3,0,4,0,..\}$ Intuitively we know that the sequence converges to $\infty$ but how do we check that rigorously. If I imitate the formal definition of convergence then I believe that we can at best come up…
Student
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Continuous function taking rationals to rationals

Is there a continuous increasing function $ f : [0, \pi] \to [0, e] $ such that $ f(0) = 0, f(\pi) = e $ and $ f (q ) \in \mathbb{Q} $ for $ q \in \mathbb{Q} $ and $ f (q ) \in \mathbb{Q}^c $ for $ q \in \mathbb{Q}^c $? I think there should be, but…
Rana
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union of two simply connected open , with open and non empty intersection in $R^2$

Let $D_1,D_2$ be two simply connected open subsets of $\mathbb{R}^2$. Let's suppose that it's intersection is nonempty and connected. Then $ D_1\cup D_2$ is simply connected. I have no idea how can I do this.
Joseph
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Prove that $ \inf_{-1 < x< 1}|f^{k}| \leq \frac{2^{k(k+1)/2}k^k}{\lambda^k} $ for $|f_{-1 < x < 1}(x)| \leq 1$ and $\lambda$ is length of $x_1 - x_0$

Let $$ |f_{-1 < x < 1}(x)| \leq 1 $$ $\lambda $ is length of conventional interval $I$ in $(-1, 1)$. Prove that on this interval: $$ \inf_{x \in I}|f^{k}| \leq \frac{2^{k(k+1)/2}k^k}{\lambda^k} $$ Where $k$ is k-th derivative, and $f^{k-1}(x)$…
Joe Half Face
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Constructing a certain rational number (Rudin)

I am not sure how this number is constructed. I know that $2>q^2>p^2$, I then also know that q>p, hence I could construct a q such that q>p. I therefore choose q=p+x, where x could be an element of of rationals and then given I know that $2>q^2$…
ALEXANDER
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Construct a set with different upper and lower Lebesgue density at zero.

For $\delta >0,$let $I(\delta)$ be the segment $(- \delta, \delta) \subset \mathbb{R}.$ Given $\alpha,\beta,$ and $0 \leq \alpha < \beta \leq 1,$ construct a measurable set $E \subset \mathbb{R}$ so that the upper and lower limits of $$m(E \cap…
Dedalus
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A bounded function in $\Bbb R$ with closed graph is continuous.

It is known that if a function $f:\Bbb R\to \Bbb R$ is continuous then its graph is closed. Proof. Let $(x_n)_{n\in\Bbb N}$ be a sequence in $\Bbb R$ so that the sequence $(x_n,f(x_n))_{n\in\Bbb N}$ is convergent in $\Bbb R^2$ at a point…
richarddedekind
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If $f$ is differentiable at $c$, then $F'$ is continuous at $c$ where $ F(x)=\int_a^x f(t)\ dt$?

Let $f$ be Riemann integrable on $[a, b]$, let $c\in(a, b)$, and let $\displaystyle F(x)=\int_a^x f(t)\ dt$, $a\le x\le b$. For the following statement, give either a proof or a counterexample: If $f$ is differentiable at $c$, then $F'$ is…
Siddhartha
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Need a hint: show that a subset $E \subset \mathbb{R}$ with no limit points is at most countable.

I'm stuck on the following real-analysis problem and could use a hint: Consider $\mathbb{R}$ with the standard metric. Let $E \subset \mathbb{R}$ be a subset which has no limit points. Show that $E$ is at most countable. I'm primarily confused…
Elements
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Uniform convergence of subsequences implying uniform convergence

Suppose ${f_n}$ are uniformly bounded and equicontinuous on some closed interval $[a,b]$. Therefore, by Arzela-Ascoli we know that $f_n$ has a uniformly convergent subsequence. But we can also apply Arzela-Ascoli to the subsequences of $f_n$ to get…
countunique
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Defining the factorial of a real number

I'm curious, how is the factorial of a real number defined? Intuitively, it should be: $x! = 0$ if $x \leq 1$ $x! = \infty$ if $x >1$ Since it would be the product of all real numbers preceding it, however, when I plug $\pi!$ into my calculator,…
Alshazgir
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