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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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Composition of a continuous function and a discontinuous function, can be continous.

Okay, I think I found an example of a continuous function $f$ composed with a discontinuous function $g$, that make a continuous function $h$. Okay let: $f:[0,1]\to [0,1)$ where $f(x)=\begin{cases}x \quad \textrm{if} \quad x\in[0,1)\\ 0 \quad…
user160110
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$\{f_n\}$ continuous such that $\inf f_n$ and $\sup f_n$ are not continuous

Find a sequence of continuous functions on $[0,1]$, $\{f_n\}$, such that $f(x):=\sup\{f_n(x):n\geq 1\}$ and $g(x):=\inf\{f_n(x):n\geq 1\}$ are both not continuous. I kept finding examples where one was discontinuous but the other was continuous…
john
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Pointwise convergence to zero, with integrals converging to a nonzero value

For $n\in{\mathbb{N}}$ let $$f_n(x)=nx(1-x^2)^n\qquad(0\le x\le 1).$$ Show that $\{f_n\}_{n=1}^\infty$ converges pointwise to $0$ on $[0,1]$. Show that $\{\int_0^1f_n\}_{n=1}^\infty$ converges to $\frac12$. I've already shown both of these…
Danny
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Rudin's definition on measurable function

In the definition of measurable function in Rudin's book, he defines measurable function from a measurable space $X$ to a topological space $Y$ as the inverse image of every open set in the range space is measurable in the domain space, the…
89085731
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properties of the integral (Rudin theorem 6.12c)

if $ f\in\mathscr{R(\alpha})$ on $[a,b]$ and if $a
Baroque Spiderman
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When doesn't a supremum exist?

Other than ∞, is there another case where a supremum (or an infimum for that matter) doesn't exist?
Grizzly0111
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Proof $\int \limits_{0}^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\frac{\pi}{2}$ by definition

Let $f(x)=\mathbb{I}_{[-t,t]}$ where $t\in (0,\pi)$. Using Parseval's theorem to this function we get: $$\sum \limits_{n=1}^{\infty}\dfrac{\sin ^2(nt)}{n^2t}=\dfrac{\pi-t}{2}.$$ Prove that $$\int \limits_{0}^{\infty}\left(\frac{\sin…
RFZ
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Can a norm only be defined on vector spaces?

All examples I have come across so far deal with norm defined on a vector space. Can norm only be defined on vector spaces?
normed
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Show that all complex Radon measures on a locally compact and $\sigma$-compact Hausdorff space is a Banach space

Let $X$ be a locally compact Hausdorff space which is also $\sigma$-compact, and let $M(X)$ be the vector space of all complex Radon measures with the total variation norm $\|\mu\|:=|\mu|(X)$. Show that $M(X)$ is a Banach space. I know that we can…
Xiang Yu
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A locally finite Borel measure on a locally and $\sigma$ compact metric space is a Radon measure

Let $X$ be a locally compact metric space which is $\sigma$-compact, and let $\mu$ is an unsigned Borel measure which is finite on every compact set. Show that $\mu$ is a Radon measure. I know that every unsigned Borel measure on a compact metric…
Xiang Yu
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How to show that the function $x^\alpha \sin\left(\frac{1}{x}\right)$ ($\alpha > 1$) is of bounded variation on $(0,1]$?

I am given the function $$f(x) = \begin{cases} x^\alpha \sin\left(\frac{1}{x}\right) &\text{on }(0,1], \\ 0 & \text{if }x = 0. \end{cases}$$ how do I show that this function is of bounded variation on $[0,1]$ if $\alpha >1$? The variation is given…
user119615
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Prove that a sequence is square summable

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty}a_nb_n < \infty$ whenever $\sum_{n=1}^{\infty}b_n^{2}< \infty.$ Prove that $\sum_n a_n^2 < \infty.$ Can anyone provide a hint to prove this ? I don't know where to start. I am…
pikachuchameleon
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Connecting the formula for the surface area of a sphere to differential forms

I have a formula for computing the area of the surface of any object in $\Bbb R^3$, namely given some parametrization $\Phi(s,t)$, I take the cross product of the partial derivative with respect to each variable and norm it. This looks suspiciously…
operatorerror
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Continuous function taking each of its values twice

I read that a continuous function can't take each of it's values exactly twice. But I don't understand why e.g. take the function $x^2$ and add one point to it very close to $0$ that also has $0$ value (and shift the rest of the function with the…
Nesa
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If $ f \in C_0^\infty$, then is $f$ uniformly continuous?

If $ f \in C_0^\infty=\{ g: g\in C^\infty, \lim_{|x|\rightarrow \infty}g(x)=0\}$, then is $f$ uniformly continuous on $\mathbb R$? ($ f : \mathbb R \to \mathbb R $)
Misaj
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