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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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Solving $x^x=\frac{1}{\sqrt 2}$

The equation $$x^x=\frac{1}{\sqrt 2},x\in \mathbb R$$ has two obvious solutions $0.5$ and $0.25$ One can easily prove they are the only ones using differential calculus. Is there any natural algebraic manipulation that would lead to finding these…
Gabriel Romon
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continuous onto function from irrationals in [0,1] onto rationals in [0,1]

Give a continuous surjective function from the irrationals in $[0,1]$ onto the rationals in $[0,1]$. Can we at least prove the existence of such a function? I couldn't see a function off the top of my head. Here we assume $[0,1]\setminus\mathbb Q$…
nk637
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Existence of non-constant continuous functions with infinitely many zeros

Possible Duplicate: A nontrivial everywhere continuous function with uncountably many roots? Does there exist a continuous non-constant real-valued function on $[a,b]$ that has infinitely many zeros? If one does exist, please give me an…
Daniel
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Harmonic function composed with conformal map is harmonic (in $\mathbb{R}^n$)

Here's the setup: Let $U,V$ open $\subset \mathbb{R}^n$, and let $u:V\rightarrow \mathbb{R}$ be harmonic, and $v:U\rightarrow V$ be conformal, i.e. $v$ is $C^1$ and the Jacobian $J_v(x)$ is a scalar multiple of an orthogonal transformation for all…
Greg O.
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Product of increasing functions is integrable in two dimensions

We say $f:[0,1]\rightarrow\mathbb{R}$ is increasing if $f(x_1)\le f(x_2)$ whenever $x_1
Paul S.
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Is this function completely monotone?

Background Long ago I bumped into an exercise in ordinary differential equations, which asks to find a solution to the differential equation: $$h'(x)=\frac{1}{2(1+xh(x))}$$ It turns out that $h(x)$ is the inverse function…
uniquesolution
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Is there a problem when defining exponential with negative base?

Well, this question may seem silly at first, but I'll make my point clear. Suppose $n \in \Bbb N$ and suppose $a \in \Bbb R$ is any number. Then the definition of $a^n$ is clear for any $a$ we choose. Indeed we define: $$a^n = \prod_{k=1}^na$$ And…
Gold
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Is a real number the limit of a Cauchy sequence, the sequence itself, a shrinking closed interval of rational numbers, or what?

I've been studying a collection of analysis books (one of them Bishop's Constructive version) and contemplating the reals. Correct me if I'm wrong, but I feel that I have seen the Cauchy sequence itself in some places and its limit in other places…
OLP
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If $\gamma_n$ are roots of $\tan x = x$ can every function be expanded in form of $\sum_n a_n \sin(\gamma_n x)$?

Solving a linear PDE, I got the general solution $$f(x,t)=\frac{e^{-t/\tau}}x\sum^\infty_{n=1}a_n\sin(\gamma_n x)$$ where $\gamma_n$ is the nth positive root of $\tan x=x$. To satisfy the initial condition we…
Szeto
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closed and open set - set $S$ is open if and only if its complement is closed?

Let set $S$ be a set of real numbers. A point $p∈S$ is set to be interior point of $S$ provided that there exist a $δ>0$ such that $(p-δ,p+δ)⊆S$. The set $S$ is said to be an open set if every element of $S$ is an interior point. How can I prove…
Diane Vanderwaif
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Help with evaluating a sum

I am trying to evaluate the following sum: $$\sum_{n=1}^{\infty} \frac{1}{n(n+1)5^n}$$ So far I have written the sum as $$\sum_{n=1}^{\infty} \frac{1}{n(n+1)5^n} = \sum_{n=1}^{\infty} \left ( \frac{1}{n} - \frac{1}{n+1} \right ) \frac{1}{5^n} =…
clarence_atwater
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Is every weak contraction a contraction?

A weak contraction is a function $f:M \to M$ such that for all $x \neq y$, $d(f(x), f(y)) < d(x, y)$. I don't think every weak contraction is a contraction, but I'm having a hard time finding a counterexample. Also, is it true that if $M$ is…
Aden Dong
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Are functions satisfying $|f(x)-f(y)|\le L \|\nabla f(x)-\nabla f(y)\|^{1+s}$ constant?

Let $f:\mathbb R^n\to \mathbb R$ be continuously differentiable. Suppose that there is $L>0,s>0$ such $$ |f(x)-f(y)|\le L \|\nabla f(x)-\nabla f(y)\|^{1+s} \quad \forall x,y\in\mathbb R^n. $$ Does this imply that $f$ is constant? Clearly if $\nabla…
daw
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What does "finite but unbounded" mean?

In the following example, Gelbaum and Olmsted (Counterexamples in Analysis, 1964, p.37) speak of a derivative being “finite but unbounded”. A function whose derivative is finite but unbounded on a closed interval. The…
user547493
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Least norm in convex set in Banach space

The following statement is true for Hilbert spaces $H$: Every closed convex set ${\cal C} \subset H$ has a unique element $x$ such that for any $y \in C$, we have $|x| \leq |y|$. Is this statement still true for Banach spaces? If not, what is a…
A wanderer
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