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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. .

145439 questions
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Continuity/differentiability at a point and in some neighbourhood of the point

For a function $f: U \to \mathbb{R}$ where $U$ is a subset of $\mathbb{R}$, it seems like that it being continuous at a point doesn't imply that there is a neighbourhood of the point where it can be continuous. Similarly, it seems like that it being…
Tim
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A power series that converges for $|x| \leq 1$ and diverges otherwise.

I need to find a power series $\sum a_n z^n$ that converges for $|x| \leq 1$ and diverges otherwise. I think I have one I just want to be sure. So, the series: $\sum \frac{z^n}{n^2}$ has radius of convergence of 1. So it converges when $|z| <1$…
user201483
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Extension of a continuous function so that the integral does not change

Suppose $f : [a,b] \to \mathbb{R}$ is a continuous function. Can another continuous function $g : [a,c] \to \mathbb{R}$ be defined such that $\int_a^b f(x)dx =\int_a^c g(x)dx$ so that $f(x) = g(x), x\in[a,b]$ and $c>b$ (Note: if $g \geq 0$ on…
jpv
  • 2,011
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f is continuous, show that f(closure) is a subset of closure of f

If $f:X\rightarrow Y$ is continuous and $E\subset X$, prove that $f(\overline{E})\subset \overline{f(E)}$. Provide an example to show that the inclusion does not have to be equality. So far what I have is that the preimage of a closed set in $Y$ is…
Struggling analyst
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Does a periodic function have to be bounded?

Let a function $f$ satisfy the relation $f(x)=f(x+1)$ for all $x\in \Bbb{R}$. Should this function always be bounded? I think so, but the book doesn't. Any help will be greatly appreciated. Please note that the function need not be continuous.
algebraically_speaking
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Does this bounded sequence converge?

The sequence $(a_n)$ is bounded for $n=1, 2, \dots$, such that $$a_n \leq \frac{1}{2} \left(a_{n-1} + a_{n+1}\right)$$ for $n \geq 2$. I want to prove the sequence $(a_n)$ converges. Since I am told the sequence is bounded, I was trying to prove it…
user186918
  • 335
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Showing $\mathbb{Q}$ does not have the L.U.B. property

I am trying to prove in a different way than how it was already proved on this website (another question). So yes, this is sort of a duplicate. Claim: $\mathbb{Q}$ does not have the least upper bound property Let $S = \{r \in \mathbb{Q} : r > 0~~…
Tyler Hilton
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Dimension score for metric spaces $\mathbb{R}^n$

Given any metric space $(X,d)$, define its score $S(X)$ to be the smallest value of $k$ such that for every $x\in X$ and $r>0$, the ball $B(x,r)$ is covered by at most $2^k$ balls of radius $r/2$. What is the growth of $S(\mathbb{R}^n)$? I guess…
simmons
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Prove if $a$ is a nonnegative real number and $n$ is a positive integer, there exists a $b \geq 0$ such that $b^n = a$

This is Theorem 7.5 in Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger. At the end of the proof in the book, we want to show by contradiction that $b^n < a$ and $b^n > a$ are not true. The proof in the book excludes the part…
mathjacks
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Can $\mathbb R$ be written as the disjoint union of (uncountably many) closed intervals?

From this post: http://terrytao.wordpress.com/2010/10/04/covering-a-non-closed-interval-by-disjoint-closed-intervals/ We know that $\mathbb R$ can't be written as the countable union of disjoint closed intervals. Can we do it if we allow uncountably…
Nishant
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2 answers

Twice differentiable function, show there is a fixed point

Let $g:[0,1] \rightarrow \mathbb{R}$ be twice differentiable with $g''(x)\gt 0$ for all $x\in[0,1]$. If $g(0)>0$ and $g(1)=1$, show that $g(d)=d$ for some point $d\in(0,1)$ if and only if $g'(1)\gt 1$. Proof. Suppose there exits $d\in(0,1)$ such…
user2467
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3 answers

If $f $ is real valued; Let $A =\{x : f(x)>x\}.$ Prove that $\sup A \in A.$

Let $f$ be a real-valued function defined on $[0,1]$ such that $f(0)>0, f(x) \ne x ~\forall x,$ and $f(x) \leq f(y)$ whenever $x \le y.$ Let $A =\{x : f(x)>x\}.$ Prove that $\sup A \in A.$ Attempt: Let us suppose that $\sup A = a \notin A.$ Then…
MathMan
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Proof of $f \in C_C(X)$ where $X$ is a metric space implies $f$ is uniformly continuous

Can you tell me if the following proof is correct? Claim: If $f$ is a continuous and compactly supported function from a metric space $X$ into $\mathbb{R}$ then $f$ is uniformly continuous. Proof: The proof is in two parts. First we want to show…
Rudy the Reindeer
  • 46,359
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1 answer

An inequality about maximal function

Consider the function on $\mathbb R$ defined by $$f(x)=\begin{cases}\frac{1}{|x|\left(\log\frac{1}{|x|}\right)^2} & |x|\le \frac{1}{2}\\ 0 & \text{otherwise}\end{cases}$$ Now suppose $f^*$ is the maximal function of $f$, then I want to show the…
molan
  • 699
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Proof that $\sum\limits_{i=1}^k \log(i)$ belongs to $O(k)$

I'm studying time complexity of binomial heaps and there's one operation (the make-heap operation) that does not make sense to me unless the following is true. $\sum\limits_{i=1}^k \log(i)$ belongs to $O(k)$ Please help me find a proof for that…
Dušan Rychnovský
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