# 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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### Limits and convolution

Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in $L^2$....
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### normed vector spaces

Suppose $A$ is a normed vector space and $a_i \in A$ and we have the following sequence $$\Vert a_1\Vert \leq \Vert a_2 \Vert \leq \cdots \leq \Vert a_n \Vert$$ for any $1\leq i<n-1$. ...
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### the Riemann integrability of inverse function

If $f \colon [a,b] \rightarrow [c,d]$ is a bijection, $f\in \mathcal{R}$ and $f^{-1}$ exists, then prove or disprove that $f^{-1} \in \mathcal{R} [c,d]$. Remark: I tried to use integration by parts ...
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### Why isn't this simpler partition enough?

I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals. Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ ...
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### An application of Fubini's theorem

Let $v_n$ be the volume of the unit ball in $\mathbb{R}^n$. Show by using Fubini's theorem that $$v_n = 2\,v_{n-1}\,\int_0^1{(1-t^2)^{(n-1)/2}\,dt}\,.$$ I've been trying to do this inductively, and ...
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### If $\sum^n_{i=1}{f(a_i)} \leq \sum^n_{i=1}{f(b_i)}$. Then $\sum^n_{i=1}{\int_0^{a_i}{f(t)dt}} \leq \sum^n_{i=1}{\int_0^{b_i}{f(t)dt}}$?

I have been puzzling over this for a few days now. (It's not homework.) Suppose $f$ is a positive, non-decreasing, continuous, integrable function. Suppose there are two finite sequences of positive ...
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### Defining dense subsets of $\mathbb{R}$

Here is a quote from Derbyshire's "Unknown Quantity" The rational numbers are "dense." This means that between any two of them you can always find another one. This being a pop. math book, this is ...
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### between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?

Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ ...
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### “On the consequences of an exact de Bruijn Function”, or “If Ramanujan had more time…”

In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely, \...