# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,472 questions
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### Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2. We now define for each $k \geq 2$ ...
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### Can the supremum of continuous functions be discontinuous at every point of an interval?

Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued ...
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### Dominated convergence 2.1?

After this question : Dominated convergence 2.0? I want to know, what about the case when $h\in L^1([0,1])$. The completed question : Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
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### Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given. ...
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### Riemann-Stieltjes integral as a limit of Riemann integrals

Let us suppose that $f, g:(A, B)\to \mathbb{R}$ are both continuous on $(A, B)$ and for $[a, b]\subset (A, B)$, suppose that $g$ is of bounded variation on $[a, b]$ (we may add, if necessary, that ...
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### Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
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### Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]
Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?