# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

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### Can a function mapping any set to its lower upper bound be bijective?

I wonder whether a function mapping any bounded set of reals A to a number that is set's least upper bound be both surjective and injective? Is it the same with function mapping to set's greatest ...
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### Is there any $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ such that $\frac{d}{dt}\|K_t^{\epsilon}\|^2\to 1$?

I am looking for conditions or at least one example such that the following holds true. Let $K_t^{\epsilon}:[0,T]\to\mathbb R$ be an smooth function such that $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ in ...
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### The definition of connected is stated as the negation of disconnected, but a little care with the logical negation, the results in a characterization. [closed]

A set E is totally disconnected if, given any two distinct points x, y ∈ E, there exist separated sets A and B with x ∈ A, y ∈ B, and E = A∪B. (a) Show that Q is totally disconnected. (b) Is the set ...
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### Prove that $\lim_{n \to \infty} \frac{\log(n+1)}{\log(n)} = 1$ [closed]

I have tried using a property of logarithm, with $$\frac{\log(n+1)}{\log(n)} = \log_{n}{(n+1)}.$$ However, I don't know how to proceed or if I'm really doing it the right way. Please, does someone ...
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### Is it possible to find the maximum of $r\ln\left(\ln\left(1+\frac{1}{r}\right)\right)$ for $r\in(0,1)$?

I was working on Chapter 5, Problem 14 of Evan's Partial Differential Equations 2nd ed and to prove integrability of $\ln \left (\ln \left (1+\dfrac{1}{|x|}\right )\right )$ on the unit $n$-ball, ...
### Approximating minimizers with $\epsilon$-nets
Let $K$ be a metric space. Fix $\epsilon > 0$. We say that $E \subset K$ is an $\epsilon$-net of $K$ if for each $x \in K$ there exists $e \in E$ such that $d(x,e) < \epsilon$. Consider a ...
### Finding functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0}$
Undergraduates at my university showed me this problem, which I found intriguing and now want to see the solution of: Find all functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0}$ such ...