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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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Inf sup of an expression not involving sequences

I have $$ \inf_{f \in W} \sup_{x \in [-1,1]} |x + f(x)|$$ where $f \in W = \{f: [-1,1] \rightarrow R$ continuous $| \int_0^1 f(x) d \mu = \int_{-1}^0 f(x) d \mu = 0\}$. I want to compute this thing but I'm not sure what to do with the $inf$ $sup$. I…
Rudy the Reindeer
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Prove that $\sup (C) = \sup (A)\sup (B)$ and $\inf (C) = \inf (A)\inf (B)$

Let $A$ and $B$ be two nonempty bounded sets of nonnegative real numbers. Define the set $C:= \{ab: a\in A, b \in B\}$. Show that $C$ is a bounded set and that $\sup (C) = \sup (A)\sup (B)$ and that $\inf (C) = \inf (A)\inf (B)$. I have asked the…
ayv2
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Proving limits using $\delta$ and $\epsilon$

I'm trying to use the $\epsilon-\delta$ definition of limits to show $$\lim_{x\to p} \sqrt{x}=\sqrt{p}$$ assuming $p>0$. I know that $|\sqrt{x}-\sqrt{p}|<\epsilon$ whenever $0<|x-p|<\delta$ Then after some algebra I get…
user101097
  • 261
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Open Dense Set of Real Numbers

I recently came across the set $O=\bigcup_{n=1}^\infty O_n$ ,where $$\mathcal{O}_n = \left(r_n - \frac{\epsilon}{2^{n + 2}}, r_n + \frac{\epsilon}{2^{n + 2}}\right)$$ where {$r_n$} is an enumeration of rationals.The above subset is dense in R and of…
happymath
  • 6,148
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1 answer

Multiple tangent function

Is there a nonlinear function $f:\mathbb{R}\to\mathbb{R}$, differentiable on $\mathbb{R}$, such that any tangent line is tangent to the graph of $f$ at two distinct points (at least)?
goon65
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What does a "weak type" inequality mean?

I saw the Hardy-Littlewood maximal inequality described as "weak-type (1, 1)". What is meant by a "weak-type" inequality in general, and what does the "(1, 1)" mean? Apparently the Marcinkiewicz interpolation theorem gives a "strong-type"…
user901823
  • 474
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2 answers

An elementary question about $\varepsilon$.

Is saying that $x\leq\varepsilon$ $\forall\varepsilon>0$ equivalent to saying $x<0$? Why? Could anyone prove it or at least guide me to prove it?
Charlie
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2 answers

Riemann integrability

Why is the Dirichlet function, $$f(x) = \begin{cases}x : x \in \mathbb{Q}, \\ 0 : x \notin \mathbb{Q}, \end{cases}$$ not Riemann integrable on any interval $[a, b]$?
user2850514
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Proving the convergence of the $p$-series without using the integral test?

I'm having trouble figuring out how to prove the convergence of the $p$-series, that is, $$\sum_{n=1}^{\infty}{\frac{1}{n^p}}$$ where $p > 1$. I'm in a real analysis course and I have a midterm coming up. I think I might need to prove this on the…
user114014
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2 answers

A problem about uniform convergence

$\{f_n\}$ are absolutely continuous functions on $[0,1]$, we know that if $f_n$ are uniformly convergent to a function $f$, then $f$ is continuous. The question is: is the function $f$ absolutely continuous?
student
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3 answers

Constructing $\mathbb R$

I am learning mathematical analysis. In one of the pages of a book on analysis I found a statement which I could not digest. The statement was "Cantor constructed $\mathbb R$ using nested intervals". Another such statement was "Dedekind constructed…
Primeczar
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Why define the integral only on measurable functions?

If $f$ is a measurable function on space $X$ with measure $\mu$, we may define $\int_X f d\mu = \sup\{\int_X s d\mu | s\text{ simple measurable on }X, s \le f\}$. But it seems that the right hand side is defined for all $f$ on $X$. So why don't we…
mr. j
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Why can't $e^{x^2}$ be integrated

My teacher told me that not only do we have to use the erf function to approximate error, but that it is proved impossible to integrate in real analysis (at least not Riemann-integrable). Is there a name for this proof, and can I have it? I am not a…
user123254
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0 answers

characteristic function of the rationals

Let $\chi$ be the characteristic function of the rational numbers in $[0,1]$. Does there exist a sequence $\{f_n\}$ of continuous functions on $[0,1]$ that converges pointwise to $\chi$?
Dave
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Generality of Lebesgue integration?

I am an electrical engineer with a strong interest in math. I took a courses in real analysis and abstract algebra 25 years ago. I tried to teach my self Lebesgue integration, and developed some level of understanding. When we consider a Riemann…
Ted Ersek
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