# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

87,792 questions
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### Prove that E=[0,1] is closed on $\mathbb{R}$. [on hold]

I know that I need to show that $E^c$=$(-\infty,0) \cup (1,+\infty)$ is open but am having a hard time figuring out how to do that. Could someone please walk me through the steps?
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### Is average operator continuous? [on hold]

Suppose $$Lf=\frac{1}{x}\int_0^xf(t)d\mu(t)$$ where $d\mu(t)=\frac{1}{t}dm(t), dm$ is the Lebesgue measure. Is it true that $L$ is continuous in $L^p((0,\infty),\mu)$, $1\leq p\leq\infty$ ?
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### What does it mean for a vector function to be twice differentiable at a point?

Let $F: R^n \to R^m$ be a vector valued function with components $F = (f_1 ,f_2 ,..., f_m).$ What is the most proper and universal accepted definition for the function $F$ be twice differentiable ...
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### Tchebyshev inequality for Measure theory.

Suppose $f$ integrable on measurable set $E$ and $f(x)$ $\geq$ $0$ on $E$, And for any $a \in \mathbb{R}^+$, we define: $E_a$ = $\{x \in E$ $\vert$ $f(x) > a\}$. (which is measurable by basic ...
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### Counterexample to Riemann sum limit

For convergent Riemann sums of $f \in C^1([0,1])$ there is the property: \tag{A}\lim_{n \to + \infty} \left[\, \sum_{k=0}^{n} f \left( \frac{k}{n+1} \right) - \sum_{k=0}^{n-1} f \left( \frac{k}{n} \...
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### Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Let $\{a_n\}_{n \geq 1}$ be a non-negative sequence of reals such that $\sum_{n \geq 1} a_n$ converges to $s$. Define $N_k = |\{n \in \mathbb{N} : a_n \geq 2^{-k}\}|$. Show that \...
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### Prove inverse of strictly monotone increasing function is continuous over the range of original function

Let $f:[a,b] \rightarrow \Bbb R$ be a strictly monotone increasing. Then $f$ has an inverse function $g:[c,d]\rightarrow \Bbb R,$ where $[c,d]$ is the range of $f$. I'm trying to prove that $g$ is ...
Let p(x) be a linear function interpolating sin(x) at $x=0, x=\frac{\pi}{2}$. Prove that $|p(x)-\sin x| \leq \frac{1}{2}(\frac{\pi}{4})^2$ on $[0, \frac{\pi}{2}]$. I've already done a bit of work and ...