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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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Prove A is open if and only if w+A is open, A is closed if and only if w+A is closed.

Let A be a subset of $\Bbb R^n$ and let $\mathbf w$ be a point in $\Bbb R^n$. The translate of A by $\mathbf w$ is denoted $\mathbf w$ + A and is defined by $$\mathbf w+A \equiv \{\mathbf w + \mathbf u\mid \mathbf u\text{ in A}\}.$$ $\mathbf a.$…
Dan C
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$\forall x \in \mathbb{R}$, there exits $\delta$ such that $(x-\delta,x+\delta) \cap A$ is countable. Prove that $A$ is countable.

As stated in the title. At the first glance I think the approach can be constructing an injection from $A$ to $\mathbb Q$, since obviously $\mathbb Q$ is a set that satisfies such condition. However I have no idea on how to get such injection. Any…
zeno tsang
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Conditional convergence, Mertens theorem

If $\sum a_n$ and $\sum b_n$ both converge and one of them absolutely then the Cauchy product $\sum c_n$ converges to $\sum a_n \sum b_n$. ($c_n = \sum_{k = 0}^n a_k b_{n - k}$), by Mertens Theorem. Now, if both converge conditionally then the…
JT_NL
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$x^x = y$, given $y$ solve for $x$ analytically

This question has been bugging me since high school where I was told "not to be concerned with such matters", but years later I still haven't found a satisfying answer. The question is really simple: $ x^x = y $ given $y$, where $y \in \mathbb{R}$…
JoelKuiper
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Prove that $f:(a, b) \to \mathbb R$ has at most countably many simple discontinuities

This is problem 17 in baby Rudin's chapter on continuity. He has a hint to use triplets of rationals that bound each simple discontinuity on the left, right, and in between the values of the limits from the left and right. It seems like this can be…
user2303321
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Prove that $2^n\alpha-[2^n\alpha]$ is dense in [0,1]

Prove that $2^n\alpha-[2^n\alpha]$ is dense in $[0,1]$, if $\alpha$ is a positive irrational number. $[x]$ represents the largest integer smaller than $x$. I only know how to prove $n\alpha-[n\alpha]$ is dense in $[0,1]$, using pigeon hole…
Doris
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Darboux's theorem of several variables

Let $f:U\longrightarrow \mathbb{R}$ a differentiable function where $U\subset\mathbb{R}^n$ open connected. What can we say about the image of the derivative $f'(U)\subset \mathbb{R}^n$? $f'(U)$ is connected? If $n=1$ , $\;f'(U)$ is an interval…
felipeuni
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A continuous function that is uniformly continuous on two sets, but not uniformly continuous on the union of these two sets?

This homework problem has just cost me 3 hours... But I still have no clue what it can be... Let $A, B \subseteq \mathbb{R}$. Find a continuous function $f:A\cup B \to \mathbb{R}$ where $f$ is uniformly continuous on $A$ and on $B$, but $f$ is not…
Qingtian
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What Does it Mean for a Function to have Finite Support?

I am working on a problem that states: Let $f$ be integrable over $\mathbb{R}$ and $\varepsilon > 0$. Show that there is a simple function $\eta$ on $\mathbb{R}$ which has finite support and $\int_{\mathbb{R}} \lvert f - \eta \rvert <…
user1876508
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Using the Mean Value Theorem to Evaluate an Integral of a Sequence of Functions?

The following statement should be true, I think, but I'm having a hell of a time trying to prove it: Let $f_n$ be a $C^1$ function on $[0,a]$, satisfying $f_n = 1$ on $[1/n,a]$ and $0\le f_n \le 1$ and $f_n(0)=0$. Let $\phi$ be a continuous function…
Braindead
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Prove that every unbounded sequence contains a monotone subsequence that diverges to inifnity.

I think I have the basic framework for this proof, but I am having trouble putting everything together in a convincing way. I plan on showing that by the definition, an unbounded sequence $(a_n)$ has infinitely many $n$ such that $a_n>M$ where…
Heath Huffman
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Given a real function $g$ satisfying certain conditions, can we construct a convex $h$ with $h \le g$?

The following is Exercise 8 from Chapter 3 of Rudin's Real and Complex Analysis (not a homework problem, just for fun). Let $g$ be a positive function on $(0, 1)$ such that $g(x) \to \infty$ as $x \to 0$. Does there exist a convex function $h$ on…
guy
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Borel set preserved by continuous map

Let $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$ be a continuous map. Show that if $A$ is a Borel subset of $\mathbb{R}^n$, then $f^{-1}(A)$ is a Borel subset of $\mathbb{R}^m$. I know that for $A$ open subset of $\mathbb{R}^n$, then $f^{-1}(A)$ is…
PJ Miller
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Sum of prime factors

If $x$ is an odd prime, is it true that the sum of the prime factors of $x + 1$ is less than $x$? If so, then this would give a nice way of constructing a "jumpy" sequence that converges to $0$, namely, let the $n$th term be $1$ divided by the sum…
user27325
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