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.

Filter by
Sorted by
Tagged with
1
vote
2answers
78 views

Is $f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y)$ Riemann-Integrable on $\mathbb{R}$?

Is $$f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y), \hspace{0.2cm} \{q_{k}\}_{k \in \mathbb{N}} \subseteq \mathbb{Q}$$ Riemann-Integrable ? ...
1
vote
2answers
61 views

Why $f_n=\frac{1}{n}1_{[0,n]}$ shows that $L^1(\mathbb R)$ is not reflexive?

In my course it's written that $f_n=\frac{1}{n}1_{[0,n]}$ shows that $L^1(\mathbb R)$ is not reflexive. Could someone explain why ? I know that reflexive mean that bounded sequence has subsequence ...
0
votes
3answers
58 views

$f'(x_0)\geq \frac{f(x_0)-f(x)}{x_0-x}$?

Assuming that $f\in\mathcal{C}^2(\mathbb{R})$ and $\max f'(x)=f'(x_0)$, then why $$ f'(x_0)\geq \frac{f(x_0)-f(x)}{x_0-x} ? $$ It might be a silly question, but I'm stuck. Thank you in advance.
0
votes
4answers
60 views

‎If ‎$‎f‎$ ‎is ‎convex ‎and ‎$‎\delta‎>0‎$‎, then ‎$‎f(x + ‎\delta‎) - f(x)‎$ ‎is ‎increasing.‎

My question is how to prove the following assertion: ‎‎If $f : I‎‎\rightarrow\mathbb{R}$ be a convex function, ‎$‎I‎$ ‎is ‎unbounded ‎above ‎and ‎‎$‎‎\delta‎>0‎$‎, then ‎$‎f_{‎\delta‎} : I‎‎\...
3
votes
3answers
109 views

Describe all the continuous function $f:\mathbb R \to \mathbb R$ such that $f \circ f=f$. Again describe all linear maps with the same property.

Can anyone describe all the continuous functions $f:\mathbb R \to \mathbb R$ such that $f \circ f=f$. I think I should search within functions whose image set is connected. ...
3
votes
1answer
63 views

How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$

How to compute $\sum_{n,m=2}^{\infty}{n^{-m}}$ Here's my progress: I suppose $\sum_{n,m=2}^{\infty}{n^{-m}}=\sum_{n=2}^{\infty}{\sum_{m=2}^{\infty}{\left(\frac{1}{n}\right)^m}}$, so we're looking at ...
0
votes
1answer
28 views

Proving a function is well defined and differentiable

Let $a, b \in \mathbb{R}^{+}_{0}$ with $a<1$ and $ab<1$. Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined as: $$f(x) = \sum_{n=0}^{\infty} a^n\cos(b^n\pi x) $$ is well defined and ...
1
vote
1answer
104 views

How to find $f(x)$ in the following integral equation?

I have the following equation: $\int_{\alpha_0}^{\alpha_1} g(\alpha)f( g(\alpha) \cdot x) d\alpha = h(x)$ where $g(\alpha)$ and $h(x)$ are both given fucntions. Specifically, $g(\alpha)=cos(\theta - \...
3
votes
1answer
44 views

Conditions for Differentiable Limit Theorem

I know that if $(f_n)$ is a sequence of differentiable functions on $(a,b)$ with pointwise limit $f$ and if $f'_n \rightarrow g$ uniformly the $f$ is differentiable on $(a,b)$ and $f' = g$. I want to ...
0
votes
1answer
37 views

A sequence $x_n$ has a cauchy subsequence ii it has a subsequence satisfying the following property [closed]

I am trying to solve this problem but could not make an idea. Please give some hint for the problem.
0
votes
2answers
57 views

Proving limits using $\varepsilon$-$\delta$ definition. How to verify that computed $\varepsilon$ definition is correct?

So from our textbook exercises I was practicing on, it says that we need to prove this limit is true using epsilon delta definition. $\lim_{(x,y)\to(0,2)}\ -3x +4y = 8$ I've came up with the result ...
3
votes
1answer
34 views

Is this “one-sided” version of the fundamental lemma of calculus of variations true?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $A:\mathbb{D}^n \to \mathbb{R}^k \otimes \mathbb{R}^k$ be smooth. Suppose that $ \langle A , V \otimes V \...
4
votes
1answer
68 views

What does “isomorphism” exactly refer to in topology?

I am reading an article and I came across the sentence that states as below: "A linear separating isomorphism from C(T) onto C(S) is continuous, in which C(S) and C(T) denote sup-normed Banach spaces ...
1
vote
1answer
61 views

Set of $x\in[0,1]$ such that 4 shows up in decimal expansion infinity often

I feel like I made a mistake, is this correct? If $$E=\{\sum_{i=1}^\infty\frac{a_i}{10^i}\in[0,1]: a_i\in\{0,1,2,\dots, 9\} \text{ and } \forall n>0,\exists m>n, a_m=4\}$$ then define $$E_n=\{\...
1
vote
1answer
44 views

Question about lim inf and lim sup

I'm looking at the proof of a property of lim inf: Given a sequence of real numbers, there is a subsequence that converges to the lim inf. For the case when the lim inf is finite, the proof in the ...
0
votes
0answers
23 views

Supremum property confirmation

Hi this is a very short question, I'm afraid this post might get deleted because it's too short. I was wondering if this is true: (in an intro analysis course) $$\forall x,y\in\mathbb{R},\ \ \sup(f(x)...
2
votes
1answer
77 views

Show $\frac{\sin(x^{3})}{x}$ is uniformly continuous on $[0,\infty)$.

Show $f(x) = \frac{\sin(x^{3})}{x}$ is uniformly continuous on $[0,\infty)$. I have the following two results to assist me: If $f$ is continuous on $(a,c)$ and u.c on $(a,b]$ and $[b,c)$, then it is ...
4
votes
1answer
101 views

Lebesgue density theorem: a martingale proof

I'm trying to prove the Lebesgue density theorem using Mantingale convergence theorem. In this post How to prove the Lebesgue density theorem using martingales? I see that someone proved it using a ...
1
vote
1answer
61 views

Let $f(x) = \sum_{r_n < x} 2^{-n}$, where $(r_n)$ is an enumeration of $\mathbb{Q}$. Why is $f$ discontinuous everywhere in $[0, 1]$?

I think I have that $f$ is discontinuous at every $q \in \mathbb{Q}$, since if $q = r_k \in (r_n)$, then we consider $(x_n) \rightarrow q$ from below and $(y_n) \rightarrow q$ from above where $x_n, ...
0
votes
1answer
43 views

Find $\alpha$ ,with the given $\sum_{n=2}^\infty\left( \lim_{n \to \infty} \frac{g_n(f_n(a))}{a!} \right)=\alpha$

For $n \in \Bbb N$, let $g_n(x)=\displaystyle x^{\frac{1}{n}}$ and $f_n(a)=\displaystyle \int_0^1(1-x^a)^n dx$. For $g$ and $f$ defined above, if $$\sum_{a=2}^\infty\left( \lim_{n \to \infty} \frac{...
1
vote
0answers
14 views

Showing the value functions for an N-player differential game solve a coupled system of parabolic PDE

I'm interested in deriving this system: $$\begin{cases}-\partial_t v^{N,i} - \sum_{j} \Delta_{x_j}v^{N,j} + \sum_{j \neq i} D_{x_j}v^{N,j} \cdot D_{x_j}v^{N,i} + \frac12 |D_{x_i}v^{N,i}|^2 = F^i(\...
0
votes
1answer
37 views

Limit question multivariable analysis

Let f be a map from $\mathbb{R}^2$ to $\mathbb{R}^2$ be defined as $f(x,y)=(x^2,y^2)$ let T from $\mathbb{R}^2$ to $\mathbb{R}^2$ be the map $T(x,y)=(2x,4y)$ then to show that f is differentiable at ...
0
votes
0answers
22 views

Is there a strictly convex function with a local non-global minimum?

have tried to prove that if the function is convex, then all locals are also global, but I don't know where I used the fact that the function is convex and not strictly convex. Suppose $f: X \subset \...
2
votes
3answers
85 views

Is $f(x,y)=\frac{1}{x^2+y^2+1}$ uniformly continuous?

Is \begin{align*} f(x,y)=\frac{1}{x^2+y^2+1} \end{align*} uniformly continuous? I was able to show that $f$ has a global maximum at $f(0,0)=1$, but I can't seem to work out a proper estimate for ...
0
votes
2answers
26 views

Prove that $g(x)=\int_a^bf(x+t)dt$ is differentiable

Let $f:\mathbb{R}\to\mathbb{R}$ be a continous function and $a,b\in\mathbb{R}, a<b$. Lets define $g(x)=\int_a^bf(x+t)dt$. Prove that $g(x)=\int_a^bf(x+t)dt$ is differentiable and that its ...
2
votes
1answer
43 views

What is the set of all values taken by $f(x) = \lfloor x \rfloor + (x - \lfloor x \rfloor) ^2$

Let $$f(x) = \lfloor x \rfloor + (x - \lfloor x \rfloor)^2$$ for all $x \in \mathbb{R}$. Then what is the set of all values taken by the function $f$? My intuition is: if we take $x$ as an integer ...
0
votes
2answers
43 views

Find a convergent power series expansion for the unique solution of $y'(x) = 1 + xy(x)$

This is a continuation of the last question I asked here: Show the associated integral operator is a contraction mapping: $x + \int_{0}^{x}ty(t)dt$ The second part asks "Find a convergent power ...
0
votes
2answers
40 views

True or false : Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$ [closed]

Statement: Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$. I'm not sure where to start with this question. Can anyone ...
1
vote
2answers
48 views

Applying the chain rule: differentiating $f(x+s)$

This might sound trivial, but I'm confused about using the chain rule on univariate real-valued functions. On Wikipedia, I see that $${\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}$$ My ...
1
vote
2answers
44 views

Formal Definition of Exponentiation (Rational Exponent)

This question has bothered me for a long time... I know some people define $(-8)^\frac{1}{3}\approx 1 +1.73i$ referring to root with the minimum argument by De Moivre's Theorem. For this question, I ...
1
vote
1answer
35 views

Can we approximate a smooth function by a continuous and nowhere smooth function uniformly?

From Stone-Weierstrass approximation theorem we know that we can approximate a continuous(no matter differentiable or not) by a polynomial function uniformly within a compact interval domain. ...
0
votes
1answer
45 views

How to calculate $I=\int_0^{\infty} \frac{dx}{x^4+a^4}$ [duplicate]

I understand it is an even function, which indicates $I=\frac{1}{2}\int_{-\infty}^{\infty} \frac{dx}{x^4+a^4}$ What should I do in the next step?
-2
votes
0answers
30 views

Integral of neither even nor odd function [closed]

We have that $M(x)$ is even function and $N(x)$ is odd function , $a$ a positive number and $b$ is a real number. $$\int_{0}^{a}e^{x}dx$$
1
vote
1answer
52 views

Let $\alpha >0$ and $ \varepsilon > 0 $. Then $\sum_{n = 0}^{ + \infty} \frac{1}{(1 + |n^2 + n - \alpha |) (1 + n + n^2)^{\varepsilon}} < + \infty $?

Let $\alpha >0$ and $\varepsilon > 0$. Then $\sum_{n = 0}^{ + \infty} \frac{1}{(1 + |n^2 + n - \alpha |) (1 + n + n^2)^{\varepsilon}} < C < + \infty $ ? where $C $ does not depends on $\...
0
votes
1answer
22 views

On the monotonicity of integrals

Let $f,g:\mathbb{R} \to \mathbb{R}$ be two functions such that $$f\left(x\right) \geq g\left(x\right), \text{ for every } x \in \mathbb{R}. \tag{1}$$ Then, by monotonicity of integrals, we have $$\...
0
votes
1answer
25 views

Minkowski sum of open and closed set

Let $A$ and $B$ nonempty subsets of $\mathbb{R}$ such that $A$ is open and $B$ is closed. Then: a) $A+B$ is open, b) $A+B$ is closed, c) $A+ int B$ is open, d) $A$ difference $B$ ...
0
votes
1answer
23 views

Taking $\lim$ out of metric $d$ [duplicate]

Let $(X,d)$ is metric space. Let $d(y_n,z_n)\to r$. And let $(y_{m_k})$ and $(z_{m_k})$ be subsequences of $(y_n)$ and $(z_n)$ respectively, with $(y_{m_k})\to y\in X$ and $(z_{m_k})\to z\in X$. (Yes, ...
0
votes
0answers
36 views

Continuous $f$ with $\int_0^1 x^n f(x) dx=0$ implies $f \equiv 0$ [duplicate]

Suppose $f$ is a continuous real-valued function on $[0,1]$ and $$ \int_{0}^{1}x^n f(x) dx=0 $$ holds for all non-negative integers $n$ How to prove that $f(x)=0$ for all $x \in [0,1]$?
0
votes
0answers
22 views

Gauss Divergence theorem vs. calculating 'on hand'

I'm having problems with the following exercise. I need to calculate a flux through a part of a unit sphere: $$ S: \{ (x, y, z) \in \mathbb{R}^3 ; x^2 + y^2 + z^2 = 1 \land x, y, z \geq 0 \} \\\ \...
1
vote
3answers
44 views

The integral $\int_{0}^{\infty} e^{-x^5}dx$ is convergent or not?

So, my question is whether the integral $\int_{0}^{\infty} e^{-x^5}dx$ is convergent or not? My work- So, I tried to use the comparison test. I see that $\frac{1}{e^{x^5}} \leq \frac{1}{x^5}$ but ...
2
votes
3answers
64 views

Solution of a differential equation hitting a constant - possible?

In my ODE lecture, we came across the DE $$\frac{dy}{dx} = y(1-y)$$ I know that the solution to this is simple to derive but there are also 2 constant solutions $y \equiv 0$ and $y \equiv 1$. I'm ...
-1
votes
1answer
33 views

If $f'(c)>f'(x)$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$, then $f'(c)>\frac{f(x)-f(c)}{x-c}$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$? [closed]

Is the above claim true? If yes, how might we prove it? If no, what is a counterexample?
1
vote
1answer
57 views

Why do they use $\frac{1}{n}$ balls in this proof for closed sets?

I can follow the proof all the way upto the fact that we need to show that no $B_\epsilon(x)$ lies completely in $T$ (thus the contradiction later). If no ball lies completely in $T$, then it must ...
-3
votes
0answers
18 views

Use the definition of the limit of a sequence to establish the following limits [closed]

1) $\lim_{n\to\infty}\frac{\sqrt n}{n+1} = 0$ 2) $\lim_{n\to\infty}\frac{(-1)^nn}{n^2 + 1} = 0$
6
votes
2answers
157 views

Prove or disprove that there does not exist a monotone function $f:\mathbb{R}\rightarrow\mathbb{Q}$ which is onto.

Prove or disprove that there does not exist a monotone function $f:\mathbb{R}\rightarrow\mathbb{Q}$ which is onto. Clearly $f$ can not be continuous. Suppose $f$ is discontinuous. Then it can have ...
-1
votes
0answers
12 views

Let f, g be defined on R and let c ∈ R, suppose that lim x->c f(x)=b, and that g is continuous at b, show that lim x-> infinity (fg) = g(b) [duplicate]

have no idea, hope for any help Let f, g be defined on R and let c ∈ R, suppose that lim x->c f(x)=b, and that g is continuous at b, show that lim x->infinity (fg) = g(b)
2
votes
2answers
60 views

The derived set $A'$ is closed by sequences

I am trying to prove that $A'$ the set of limit points is closed using different methods. For this I am using sequences and I wrote two answers that I don't know if it is right. Answer 1 Let $(x_n) \...
2
votes
2answers
29 views

Finding appropriate $\delta$ to prove continuity

I want to show that $f(x) = \sqrt{x^2 + 5}$ is continuous at $x = \pi$ using the epsilon delta definition. Here's how far I could go: $|x - \pi| < \delta$ is same as $|x^2 - \pi^2| < \delta|x+\...
2
votes
1answer
46 views

Prove that $\sum_{n=1}^{\infty} \frac{z^2}{n^2 - z^2 }$ converges uniformly

I'm trying to show that the serie $\sum_{n=1}^{\infty} \frac{z^2}{n^2 - z^2 }$ converge uniformly over all compact subsets of $B(0,1)$. If we take a compact $K\subset B(0,1)$ for all $z\in K$ we have $...
2
votes
2answers
55 views

$f, g: \mathbb{R} \to \mathbb{R}$ and $f(x+h) = f(x) + g(x)h + a(x,h)$ for $|a(x,h)| \leq Ch^3$. Show that $f$ is affine.

Let $f, g: \mathbb{R} \to \mathbb{R}$ be functions that obey $f(x+h) = f(x) + g(x)h + a(x,h)$ for $|a(x,h)| \leq Ch^3$ for all $x, h \in \mathbb{R}$ and for some constant $C$. Show that $f$ is affine (...