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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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For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$

Suppose $f$ is a Riemann integrable function on $[0,1]$. Prove that $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$ This is what I am thinking: Fix $n$. Then by Jensen's Inequality we have $$0\leq\left(\int_0^1x^nf(x)dx\right)^2 \leq…
Galois
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Limit of $\frac{f(x)}{x}$ when $f'(x)$ tends to infinity

While solving a problem, I found the statement below If $f:[0,+\infty)\rightarrow \mathbb{R}$ and $f$ is differentiable on $(0,+\infty)$, then $\displaystyle \lim_{x\rightarrow +\infty} f'(x)=+\infty$ if and only if $\displaystyle…
Tien Kha Pham
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Irrational algebraic numbers

A number is rational iff its binary expansion is repeating. My main question is: What restrictions does the property of beeing an irrational, but algebraic number place on its binary expansion? Can we prove that for every irrational algebraic number…
TROLLHUNTER
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Monotone increasing function satisfying FTC is absolutely continuous?

Let $f:[0,1] \to \mathbb{R}$ be a monotone increasing function so that $f(0)=0=\lim_{x\to{0^{+}}}f(x)=f(0)$ and $f(1)=1=\lim_{x\to{1^-}}f(x).$ If $\int_0^1f'(t)dt=1$ show that $\int_a^b f'(t)dt=f(b) - f(a)$ for all $0 \leq a < b \leq 1$ and that…
Darrin
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Continuously differentiable map from $\mathbb{R}^{m+n}$ to $\mathbb{R}^n$

Suppose $m,n>0$, $U$ an open subset of $\mathbb{R}^{m+n}$ and let $f: U \to \mathbb{R}^n$ be continuously differentiable. Is it possible for $f$ to be injective? My thinking is that continuous differentiability+injectivity suggests that this is…
goatman2743
  • 549
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Maximum $\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$ where $y\in \left[0,1\right]$?

How find maximum this integral $$\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$$ where $y\in \left[0,1\right]$?
piteer
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Is every continuous function that preserves (ir)rationality a rational function?

Let $f: [a, b]\to \mathbb R$ be a continuous function such that for all $r\in\mathbb R$, $f(r)$ is rational iff $r$ is rational. Does it follow that $f$ is a rational function?
Nishant
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Is this two-dimensional version of the Intermediate Value Theorem correct?

Given 1 continuous function $f(x)$ defined on a 1-dimensional interval $[-1,1]$, the IVT says that if: $f(-1)<0$ and $f(1)>0$ then there is an $x$ such that $f(x)=0$. I am trying to prove the following extension: Given 2 continuous functions…
Erel Segal-Halevi
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Does there exists a measure $\mu$ on $[0,1]$ such that $\int_0^1 p(x) \, d\mu(x) = p'(0)$ for every polynomial $p$

If this polynomial is at most degree $n$, I know this measure exists, but I am not sure that whether there exists $\mu$ for every polynomial?
noname1014
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Make new countable sets from old ones.

Here is the proof from my lecture notes that if you have two countable sets $A$ and $B$, you can make another countable set $A \times B$ from them. Let $A$ and $B$ be countable sets. Let $f : A \rightarrow \mathbb{N}$ and $g : B \rightarrow…
Elise
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Upper Riemann Sum of the Thomae function

I'm reading a real analysis text, and the author want to show that the upper Riemann sum of the following function is always $\lt \epsilon $ : $f:[0,1] \longrightarrow \mathbb{R}$ given by $f(x) = 0$ if $x \notin \mathbb{Q}$ and $f(x)=\frac{1}{q}$…
Jr.
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consequence of mean value theorem?

Suppose that $f:[0,1]\to\mathbb{R}$ is differentiable on (0,1) and continuous on [0,1]. Would like to assert that if f(0)=0, and $|f'(x)|\leq |f(x)|$ for each $x\in (0,1)$, then f is the zero function. I have tried applying several different…
reluctant mathematician
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Proving Riemann integrability using sequences of Riemann sums

I am trying to prove the following: Suppose $ f:[a,b]\rightarrow\mathbb{R} $ is bounded. Then $ f $ is Riemann integrable if and only if for each sequence of marked partitions $\{P_n\}$ with $\{\mu(P_n)\}\rightarrow0$, the sequence $\{S(P_n,f)\}$…
Shafat Arbaz Alam
  • 287
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Function differentiable on $(a,b)$ but not continuous on $ [a,b]$

Is there any function $f$ which is differentiable on an open interval $(a,b)$ but is not continuous on (and also cannot be extended continuously to) the closed interval $[a,b]$?
Anne
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Implicit Function Theorem and sufficient vs. necessary conditions

I have a question regarding the Implicit Function Theorem which I'll ask by way of an example... Can the equation $\sqrt{x^2+y^2+2z^2}=\cos(z)$ be solved uniquely for $y$ in terms of $x$ and $z$ near $(0,1,0)$? For $z$ in terms of $x$ and…
mrmingus
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