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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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Partition of unity on locally compact Hausdorff space

Let $X$ be a locally compact Hausdorff space, and let $(U_\alpha)_{\alpha \in A}$ be an open cover of $X$. Show that there exist compactly supported continuous functions $f_\alpha: X \to [0,1]$ supported on $U_\alpha$ for each $\alpha \in A$ with…
Xiang Yu
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Why is the dual of $L^\infty$ not $L^1$?

From the Riesz representation theorem, we know that the dual of $L^p$ is isomorphic to $L^q$ for $1
Xiang Yu
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An example of an algebraically open set in $\mathbb{R}^2$ which is not open?

Definition. A subset $U$ of a real vector space $V$ is algebraically open if the sets $\{t\in\mathbb{R}:x+tv\in U\}$ are open for all $x,v\in V$. In the real vector space $\mathbb{R}^2$ equipped with the usual topology, it is clear that every…
Xiang Yu
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Proving a function is not Riemann integrable

Prove that the bounded function $f$ defined by $f(x)=0$ if $x$ is irrational and $f(x)=1$ if $x$ is rational is not Riemann integrable on $[0,1]$. I was given the hint to use the inverse definition of Riemann integrable and consider the cases of the…
xCanaan
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Hahn decomposition for signed measure not unique (Folland)

In Folland's Real Analysis, p.87, The decomposition $X = P \cup N$ if $X$ as the disjoint union of a positive set and a negative set is called a Hahn decomposition for $\nu$. It is usually not unique ($\nu$-null sets can be transferred from $P$ to…
sleeve chen
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Sequence of differentiable functions

Let $f_n$ be a sequence of differentaible functions on $[0,1]$ to $\mathbb{R}$ converging uniformly to a function $f$ on $[0,1]$, Then $f$ is differentiable and Riemann integrable there $f$ is uniformly continuous and R-integrable $f$ is…
Myshkin
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A few questions on the different aspects of differentiation from $\mathbb{R}^2 \to \mathbb{R}$

Given the function $$f(x,y) = \begin{cases} \frac{x^3y^2}{x^4 + y^4} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ How would you prove (or disprove) the following statements: $f$ has all partial derivatives at $(0,0)$. $f$ has all…
user26069
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Check whether sets are open, closed, compact or bounded

For each of following sets decide whether it is open, closed, compact or bounded: (a) $\left\{ (x,y)\in\mathbb{R}^2 : \sin\left( \sin\left( \cos(xy)\right)\right)=\sin\left(\sin(x+y) \right)\cdot y , \ x\ge -1 , \ y\le x \right\}$ (b) $\left\{…
xan
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If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, $f(x) \neq f'(x)$, show that $\{x\in [0,1] \text{ and } f(x) = 0\}$ is finite.

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, $f(x) \neq f'(x)$ for all $x$, show that $\{x\in [0,1] \text{ and } f(x) = 0\}$ is finite. I have shown that there cannot be an interval $[a,b]$ contained in $[0,1]$ such that $f(x) = 0$…
continental
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Riemann integral of continuous non-negative function

Suppose $f\geqslant 0$, $f$ is continuous on $[a,b]$, and $\int \limits_{a}^{b}f(x)dx=0.$ Prove that $f\equiv 0$ on $[a,b]$. Proof: We'll define function $F(x)=\int \limits_{a}^{x}f(t)dt$. We know that $F(x)$ is continuous since $f$ is bounded and…
RFZ
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Function bounded below, with negative 2nd derivative. Show function is constant.

$f:\mathbf{R} \rightarrow \mathbf{R}$ is twice differentiable. $f''(x) \leq 0$ $\forall x \in \mathbf{R}$. $f$ is also bounded below. Show $f$ is a constant function. I've got to $$f(x+y)-f(x) \leq f(x)-f(x-y)\, \forall x \in \mathbf{R}, y>0$$…
Amy
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Integrals conditioning relations

Let be $f:[0,1]\to\mathbb R$ a continuous function such that: $$\int_0^1 f(x) dx = 0 $$ Prove that there exists $c\in(0,1)$ such that: $$\int_0^c xf(x) dx = 0 $$ I tried to go the integration by parts way but got stuck. I'm also…
user 1591719
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Showing that $\lim\sup s_n = \lim\inf s_n =s$

I have the following problem: Let $s$ be a real number and $(s_n)$ be a sequence of real numbers. Suppose that for any subsequence $(s_{n_{k}})$ of $(s_n)$, $(s_{n_{k}})$ has a subsequence $(s_{n_{k_{l}}}$) satisfying $$ \lim_{l \to \infty}…
user284639
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Find an open cover of the interval $(-1,1)$ that has no finite subcover

Find an open cover of the interval $(-1,1)$ that has no finite subcover. I have no idea how to do this problem. Can someone help me go step by step until I get this concept? Thanks you!
ematth7
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$f : (0,1) \rightarrow \mathbb{R}$ countinous with non-negative right-hand derivative is non-decreasing

Let $f : (0,1) \rightarrow \mathbb{R}$ continous, right-hand derivable, such that $f'_+ \geqslant 0$ $\forall t \in (0,1)$. Show that $f$ is non-decreasing. First I thought of Lagrange's theorem but I don't think that's useful here. Then, for every…
un umile appassionato
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