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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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What is the 0th root-of-mean-of-powers?

On $\mathbb{R}^n$ and $p\ge 1$ the $p$-norm is defined as $$\|x\|_p=\left ( \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$ and there is the $\infty$-norm which is $\|x\|_\infty=\max _j |x_j|$. It's called the $\infty$ norm because it is the limit of…
Florian
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Sequence of monotone functions converging to a continuous limit, is the convergence uniform?

I'm reading some extreme value theory and in particular regular variation in Resnick's 1987 book Extreme Values, Regular Variation, and Point Processes, and several times he has claimed uniform convergence of a sequence of functions because…
BlueBuck
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$f^2+(1+f')^2\leq 1 \implies f=0$

Find all $f\in C^1(\mathbb R,\mathbb R)$ such that $f^2+(1+f')^2\leq 1$ It's quite likely the answer is $f=0$. Note that $|f|\leq 1$ and $-2\leq f'\leq 0$. Therefore $f$ is decreasing and bounded. What then ? I tried contradiction, without…
Gabriel Romon
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Intuition for the Universal Chord Theorem

So the Universal Chord theorem is a statement and proof that; The numbers of the form $r = \displaystyle \frac{1}{n} \ \ n \ge 1$ are the only numbers such that for any continuous function $\displaystyle f:[0,1] \to \mathbb{R}$ such that…
pad
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Locally Lipschitz implies continuity. Does the converse implication hold?

Let $A$ be open in $\mathbb{R}^m$; let $g:A\rightarrow\mathbb{R}^n$. If $S\subseteq A$, we say that $S$ satisfies the Lipschitz condition on $S$ if the function $\lambda(x,y)=|g(x)-g(y)|/|x-y|$ is bounded for $x\neq y\in S$. We say that $g$ is…
Mika H.
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If $f$ is uniformly differentiable then $f '$ is uniformly continuous?

The following theorem is true? Theorem. Let $U\subset \mathbb{R}^m$ (open set) and $f:U\longrightarrow \mathbb{R}^n$ a differentiable function. If $f$ is uniformly differentiable $ \Longrightarrow$ $f':U\longrightarrow…
felipeuni
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Necessary and Sufficient Conditions for Riemann Integrability

A function is called Riemann integrable if and only if it is bounded and continuous almost everywhere on its domain. However, I have read that the following two statements are also true: a) If $f$ is continuous then $f$ is Riemann integrable b) If…
user71284
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Show that $\mathbb{Q}$ is dense in the real numbers. (Using Supremum)

I am stuck on a homework the teacher gave us to hand in. It is stated as following: The set of rational numbers $\mathbb{Q}$ given by all $q = \frac{m}{n}$ for some $m, n \in \mathbb{Z}$ with $n \neq 0$ is dense in the real numbers in the following…
padrino
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Derivative of a linear transformation.

We define derivatives of functions as linear transformations of $R^n \to R^m$. Now talking about the derivative of such linear transformation , as we know if $x \in R^n$ , then $A(x+h)-A(x)=A(h)$, because of linearity of $A$, which implies that…
Theorem
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Partial sums of exponential series

What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$? Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because we're cutting off the series right at the largest…
hmakholm left over Monica
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5 answers

Find the limit points of the set $\{ \frac{1}{n} +\frac{1}{m} \mid n , m = 1,2,3,\dots \}$

I need to find limit points of the set $\{ \frac{1}{n} +\frac{1}{m} \mid n, m = 1,2,3,\dots \}$. My try : If both $m$ and $n$ tend to very large values say $\infty$ then the value of $\{ \frac{1}{n} +\frac{1}{m} \}$ tends to $0$, and if only one…
Rusty
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Product of two Lebesgue integrable functions not Lebesgue integrable

I have a homework problem that says; Give Borel functions $f,g: \mathbb{R} \to \mathbb{R}$ that are Lebesgue integrable, but are such that $fg$ is not Lebesgue integrable. I saw this page too: Product of two Lebesgue integrable functions, but the…
nate
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Explanation of the Bounded Convergence Theorem

Theorem (Bounded Convergence Theorem) Let $\{f_n\}$ be a sequence of measurable functions on a set of finite measure $E$. Suppose $\{f_n\}$ is uniformly pointwise bounded on $E$, that is , there is a number $M\geq 0$ for which $|f_n| \leq M$ for…
emka
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5 answers

Continuous function from $(0,1)$ onto $[0,1]$

While revising, I came across this question(s): A) Is there a continuous function from $(0,1)$ onto $[0,1]$? B) Is there a continuous one-to-one function from $(0,1)$ onto $[0,1]$? (clarification: one-to-one is taken as a synonym for injective) I…
yoyostein
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3 answers

Characterization of sets of differentiability

If $f : \mathbb{R} \to \mathbb{R}$, define $C(f) = \{ x : f \text{ is continuous at } x \}$ and $D(f) = \{ x : f \text{ is differentiable at } x \}$. I have seen it proved that: $C(f)$ is a $G_\delta$ set. For any $G_\delta$ set $A \subset…
Ian
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