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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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Proving That Multiplication among Interval Numbers is Associative

I'm currently working on a problem where I need to provide a rigorous proof that multiplication among interval numbers is a associative. For those of you who haven't heard of an interval number before, here are some definitions: We say that $x$ is…
Elements
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If $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with $|f|$ uniformly continuous, is $f$ necessarily uniformly continuous?

I don't think so. But, I can't find the counterexample. The singularity shall be at infinity because $f$ is uniformly continuous over any bounded and closed intervals $[a,b]$. Also, $f$ must oscillate at infinity. Otherwise, the uniform continuity…
Yuhang
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Introduction to Real Analysis books

I am looking for a book that covers introduction to real analysis. Currently, I am reading The Elements of Real Analysis, by Robert Bartle. However, I quickly noticed that about half of the theorems and all of the sample questions don't have…
Robben
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Prove that there does not exist a strictly increasing function satisfying $f(2)=3$ and $f(a\cdot b)=f(a)\cdot f(b)$

I stumbled upon the following problem, but I can't seem to find a way to prove it. Show that there does not exist a strictly increasing function that maps the set of natural numbers to natural numbers, satisfying $f(2)=3$ and $f(a\cdot b)=f(a)\cdot…
DMH16
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If $f(x+y)=f(x)\cdot f(y)$, then $f(x)=e^{ax}$

I'm trying to prove this question: Let $f:\mathbb R\to \mathbb R$ a function such that $f(x+y)=f(x)\cdot f(y)$, $f(0)=1$ and $f'(0)=a$. Show that $f(x)=e^{ax}$, for every $x\in \mathbb R$. First of all I'm trying to prove that this function is…
user42912
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Uniform continuity by passing to the extended real line

Assume that $f : \mathbb{R} \to \mathbb{R}$ is continuous and $\lim_{x \to \pm \infty} f(x)$ exists. Can we argue that $f$ is uniformly continuous, by saying that a continuous extension of $f$ to the extended real-line (two-point compactification)…
passerby51
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Order preserving bijection from a countable subset of R to a subset of N

Let A be a countable subset of $\Bbb R$ which is well-ordered with respect to the usual ordering on $\Bbb R$ (where ‘well-ordered’ means that every nonempty subset has a minimum element in it). Then A has an order preserving bijection with a subset…
Cloud JR K
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When do a function's tangent lines touch every point?

Consider a function $f : \mathbb{R}\rightarrow\mathbb{R}.$ Say that $f$ is big if every point of the plane lies on the graph of $f$ or a tangent line to $f.$ What can we say about big functions? Is there already a name for this property? I've got…
user147556
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Bijective function excluding 0

Construct a bijective function $f:\mathbb R\to\mathbb R\setminus\{0\}$. Prove that the function is bijective. Im having trouble with this... A few concepts that I get so far is that function essentially should be representing $\mathbb R$, which in…
sphynx888
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Show pairwise disjoint open intervals at most countable

I am completely lost in this proof. Can anyone please help me out? Statement to prove : Show that any collection of pairwise disjoint, nonempty open intervals in $R$ is at most countable, Hint : each one contains a rational Proof: Let {$g_n$} be…
Frank
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Approximation of every continuous function on $[a, b]$ by polynomials from countable set of polynomials

Is there a countable subset of polynomials $C$ with the property that every continuous function on $[a, b]$ can be uniformly approximated by polynomials from $C$? This is problem from Abbott Understanding Analysis. I know that there exists a…
Hash Nuke
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Almost everywhere (ae) Homogeneous function of degree $0$ equals to a constant for almost every $x \in (0,\infty)$?

Is an almost everywhere (ae) Homogeneous function of degree $0$ equals to a constant for almost every $x \in (0,\infty)$ ? Let $f : \mathbb R \to \mathbb R$. If $f(ax)=f(x)$ ae for any $a>0$ Then $f(x)=c$ for almost every $x \in (0,\infty)$,…
ibnAbu
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find the value of L?

Suppose f is continuous on $[0,1]$. Find $$L=\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} (-1)^k f(\frac{k}{n}) $$ My attempts : I was thinking about Riemann sum... but $(-1)^k$ creates confusion. Thanks in advance.
jasmine
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How to prove $|f(x)|\leq \frac{3}{2}$ for all $x\in [-1, 1]$

Let $f(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. Suppose $f(-1),f(0), f(1) \in[-1, 1]$. Prove that $|f(x)|\leq \frac{3}{2}$ for all $x\in [-1, 1]$. Here I need to show $|f(x)|\leq \frac{3}{2}$. It means $f(x)$ lies between $-3/2$ and…
A.D
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How to prove that $\pi=\pi$?

I am trying to prove that indeed $\pi=\pi$. More precisely, that: $$6\sum_{n=0}^\infty \frac{(2n)!}{4^n (n!)^2 (2n+1) 2^{2n+1}}=\pi$$ Where the definition of $\pi:$ $$\pi=4\sum_{n=1}^\infty\frac{(-1)^{(n+1)}}{(2n-1)}$$ Using the epsilon delta…
Dole
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