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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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Determining the Length of a Curve Using Partitions

I have encountered the following problem: Let $f$ be continuous on $[a,b]$. Define the length of $f$ on $[a,b]$ by $$l=\sup_P[\lambda_P(f)],$$ where $$\lambda_P(f)=\sum_{k=1}^N\sqrt{(x_k-x_{k-1})^2+(f(x_k)-f(x_{k-1}))^2},$$ and the supremum…
wjmolina
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Solving $f'(x) = f(x+1) - f(x)$

Find all $f \in \mathscr{C^\infty}(\mathbb R)$ that satisfy the equation $$f'(x) = f(x+1) - f(x).$$ The 'obvious' answer is the set of all affine maps, but I'm not entirely sure. Some progress: For any $x \in \mathbb{R}$, we have $$f(x+h) =…
user217285
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Prove $\sum na_n$ converge if $\sum (a-s_n)$ converge

Let $\sum a_n=a$ with terms non-negative. Let $ s_n$ the n-nth partial sum. Prove $\sum na_n$ converge if $\sum (a-s_n)$ converge
El Chapo
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Show that $x \leq f(x) \leq 2x, \forall x\geq0$

Prove: $$x \leq f(x) \leq 2x, \forall x\geq0$$ conditions: $f$ is differentiable $f(0) = 0$ $1 \leq f'(x) \le 2, \forall x\ge0$ I've tried to do it by limit defn but couldn't seem to get to the right solution: $$ 1 \le \lim_{x \to c}…
adsisco
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Prove that $|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$

Question: Using the Mean Value Theorem, prove that $$|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$$ for all $a,b∈(1/2,1)$. Here, $\sin^{−1}$ denotes the inverse of the sine function. Attempt: I think I know how to do this but I want to make sure that I am as…
cooldudeman
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Lebesgue Decomposition Theorem

The usual statement of the Lebesgue Decomposition Theorem says that given two $\sigma$-finite measures $\mu$ and $\nu$ on a measure space, we can decompose $\nu = \nu_1 + \nu_2$, where $\nu_1$ is absolutely continuous with respect to $\mu$ and…
user1736
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Intuition behind uniformly continuous functions

I'm trying to have an intuition behind the uniformly continuous functions. Something to show to my students. For example, before giving the formal definition and some examples of continuous functions, we can say to beginners that roughly speaking,…
user42912
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Pointwise convergent and total variation

I'm preparing for a test for real analysis and I came across this problem in Royden's book: Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges pointwisely on $[a,b]$ to the real valued function $f$. Show that $TV(f) \leq…
Vokram
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Proving that a set isn't dense

Let $A$ be a set of real numbers such that $A \subseteq [0,1]$. I'm having a hard time proving that $C=\left\{\frac{a+1}{n^2} \colon a \in A, n \in \mathbb{N} \right\}$ is not dense in $[0,1]$. How should I approach this? I know that I must find an…
user225402
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Proof of squeeze theorem for functions

Suppose for all $x$ we know $g(x)\le f(x)\le h(x)$ and $\lim_{x\to c} g(x)=L=\lim_{x\to c} h(x)$. Does the following argument work to conclude that $\lim_{x\to c} f(x)=L$? Let $\epsilon\gt 0$ be given. Then we can find a $\delta_1$ such that if…
Exit path
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$f(x)$ is non-negative and $ \int_a^bf(x)dx = 1 $, show that $ [\int_a^bf(x)\cos{kx}dx]^2 + [\int_a^bf(x)\sin{kx}dx]^2 \leq 1 $

Suppose $f(x)$ is non-negative and integrable on $[a, b]$, and that $ \int_a^bf(x)dx = 1 $. Prove that $$ [\int_a^bf(x)\cos{kx}dx]^2 + [\int_a^bf(x)\sin{kx}dx]^2 \leq 1.$$ Thanks! There is a hint that the problem has something to do with the…
leafpile
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Show $\int_0^\infty x^{-\alpha}\sin x dx$ exists for $\alpha \in (0,2)$.

Show that $\int_0^\infty x^{-\alpha}\sin x dx$ exists for $\alpha \in (0,2)$. This is a real analysis class question. I am not quite sure how to show this. I tried a whole bunch of things like integration by parts, bounding sine, ... and nothing…
Galois
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Prove $d(x,y)=\arctan|x-y|$ is a metric

I have to show that $d$ is a metric on the real numbers, and the first three axioms are straight forward, the triangle inequality poses a problem. I know we need to get $$ \begin{align*} d(x,y) &= \arctan|x-y| \\ &\le \arctan|x-z|+\arctan|z-y|…
Jeff
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Soft question: Union of infinitely many closed sets

this is a question that is not addressed in my book directly but I was curious. We just proved that the union of a finite collection of closed sets is also closed, but I was curious about if the union of infinitely many closed sets can be open. This…
user1236
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Proving that there is an irrational number between any two unequal rational numbers.

I'm trying to prove that there is an irrational number between any two unequal rational numbers $a, b$. Here's a "proof" I have right now, but I'm not sure if it works. Let $a, b$ be two unequal rational numbers and, WLOG, let $a < b$. Suppose to…
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