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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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Is there any two variable function which has no representation of the form $\sum\limits_{n=1}^{\infty} f_n(x)g_n(y)$?

Is there any function$f:\Bbb R^2\to\Bbb R$ which has no representation of the form below? $$f(x,y)=\sum_{n=1}^{\infty} g_n(x)h_n(y)\quad(x,y\in \Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find…
Zeinab
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Intuition for $\lim\sup$ and $\lim\inf$

After reading several alternative definitions of $\lim\sup$ and $\lim\inf$, such as $\lim\sup$ being the supremum of the set of all subsequential limits, I'm still having trouble building the intuition for $\lim\sup$. One thing that I feel is true,…
gws
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Prove that the given set is bounded above in $\mathbb{Q}$ but does not have a Supremum in $\mathbb{Q}$

Let $S=\{r\in \mathbb{Q}\mid r\leq\sqrt{3}\}$. Now prove that $S$ is bounded above in $\mathbb{Q}$ but it does not have a supremum in $\mathbb{Q}$. The following is the proof I came up with, but I do not feel confident with it. Please let me know…
enlgmatlc
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If $x_0=x$ and $x_{i+1}=x_i+\sqrt{x_i}$, in how many steps $x_r \ge 2x$?

In the question of the title, if I define $r(x)=\inf\{s\in \mathbb {N}:x_s\ge 2x\} $, i want to know how $r(x)$ behaves asymptotically, more precisely to know if $\lim\limits_{x\rightarrow \infty} \dfrac {r(x)}{\sqrt{x}}$ exists, and in that case,to…
lecheconmilo
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Non-negative concave functions are non-decreasing

Let $f : [0, \infty) \longrightarrow [0, \infty)$ be concave with $f (0) = 0$ and $f (x) > 0$ for $x > 0$. Show that $f$ is non-decreasing. It is clear that a concave function $f$ can be decreasing. Intuitively, $f$ must then become negative beyond…
Andy
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I want to prove that the following set is closed

Let $A\subseteq R$ be a compact set and $B\subseteq R$ closed. Then $S=\{b\sin a;b\in B,a\in A\}$ is closed. What I have done is to consider the continuous function $$f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$$ defined by $f(x,y)=x\sin y.\;$…
Aliakbar
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An example of a bounded countably infinite subset of the real numbers.

I've been trying to think of an example of bounded, countably infinite subset of the real numbers. However, knowing that countably infinite means can be put into 1-1 correspondence with the naturals, this doesn't seem intuitively obvious. Thanks in…
user55511
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Fixed point of a monotone on [0,1].

prove or disprove: Let $f:[0,1]\rightarrow[0,1]$ be a monotone (need not be strict) function then f has a fixed point. Can I have a hint?
Selvakumar A
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Cute undergraduate real analysis problem

Possible Duplicate: Showing range is countable I was going through my old notebooks and found this. I thought it was good enough to share. Problem: Let $f(x)$ be a function $\mathbb R \rightarrow \mathbb R$, with the only restriction on it being…
Potato
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Finding upper and lower derivatives

Let $$D^+f(x)=\lim\limits_{h \to 0}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right] \text{ and } D^-f(x)=\lim\limits_{h \to 0}\left[\inf\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$$ represent the upper and lower derivative…
emka
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For function $f:[0,1]\rightarrow \Bbb R$ we know : $f(x+y)\geq f(x)+f(y)$.Prove that $f(x)\leq 2x$

Function $f:[0,1]\rightarrow \Bbb R$ holds in the following conditions : 1) $f(1)=1$ 2) $\forall x \in [0,1]: f(x)\geq 0$ 3) $\forall x,y,x+y \in [0,1]: f(x+y)\geq f(x)+f(y)$. Prove that $\forall x \in [0,1] : f(x)\leq 2x$ I tested…
Hamid Reza Ebrahimi
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Questions about derivative and differentiation

For a real-valued function $f$ defined on $\mathbb{R}$ or its subset, is it possible that it is differentiable at one point and not in one of its neighbourhoods except the point itself? is it possible that it is differentiable over an interval,…
Tim
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How to prove that the image of a continuous curve in $\mathbb{R}^2$ has measure $0$?

How to prove that the image of a continuous curve in $R^2$ has measure $0$? This is an exercise given in Real Analysis-Stein & Shakarchi. A hint is given as follow: Cover the curve by rectangles, using the uniformly continuity of $f$.
Sleepingip
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Sequences of simple functions converging to $f$

Proposition Let $f$ be a bounded measurable function on $E$. Show that there are $\{\phi_n(x)\}$ and $\{\psi_n(x)\}$ - sequences of simple functions on $E$ such that $\{\phi_n(x)\}$ is increasing and $\{\psi_n(x)\}$ is decreasing and each of…
emka
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Possible alternate proof of uniqueness of power series?

I want to show that if $\sum_{k=0}^\infty a_kx^k = 0$ on $[0,1]$, then $a_k=0 \forall k\in\mathbb{N}$. I'm aware of the standard proof, but wanted to try another argument. We know that a polynomial of degree $k$ has at most k roots, so the…
Jan Lynn
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