# Tagged Questions

Theoretical foundations of calculus: 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.

880 views

### Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
56 views

102 views

### What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
41 views

68 views

### Evaluation of Real Integral

Given the following definition:$$I=\int\limits_{0}^{2\pi}e^{-i\theta n}\left(\frac{1}{n}\right)^{\rho e^{i\theta}}d\theta$$ Is there an analytic method for evaluating this integral? Best Regards
13 views

### Lipschitz method writing the unique solution.

So the problem gives $f(t,y) = y \cos t$ with $t$ between or equal to $0$ and $2$. I already know the lipschitz method holds with $L=1$. But I'm not sure how to find the unique solution which turned ...
47 views

### Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
120 views

58 views

### Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
26 views

### Numerical differentiation (approximation with three supporting points )

Given the supporting points $x-2h,x-h,x+2h$. Determine the difference quotient Du(x) in the form $$Du(x)=au(x-2h)+bu(x-h)+cu(x+2h)$$ for the numerical approximation of $u'(x)$ of order $2$. What ...
73 views

### Use Fourier Transform to Show that $f=0$ a.e.

I was working through an old qualifier on my own when I ran across this following question that I was unable to crack. Here it is verbatim: "Let $f\in L^2(\mathbb{R}, \mathcal{L}, m)$ and suppose ...
80 views

54 views

77 views

### Determine the value of $p$ for which the following infinite series converges and for which it diverges.

Determine the value of $p$ for which the following infinite series converges and for which it diverges: $$\sum_{n = 2}^{\infty} \frac{\sqrt{n + 2} - \sqrt{n - 2}}{n^{p}}.$$ I don’t know how to ...
164 views

### Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test it'...
132 views

### Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
117 views

### Determine if this series $\sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges

Determine if the following series converges: $$\sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}.$$ (http://i.stack.imgur.com/qWiuy.png) I don't know how to start.
91 views

### Does this function achieve a maximum or minimum?

Suppose that $u$: $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ is continuous and $\lim_{x\to\infty}$ $u(x)$=$\lim_{x\to-\infty}$ $u(x)$=0. Does $u$ achieve a maximum or a minimum value on $\mathbb{R}$? I ...
51 views

### Determine if $\sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}$ congerges

Determine if the following series converges: $$\sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}.$$ I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
388 views

230 views

### How can I prove the integral $\int_{1}^{x} \frac{1}{t} \, dt$ is $\ln x$ with this approach?

I have been trying to find a proof for the integral of $\int_1^x \dfrac{1}{t} \,dt$ being equal to $\ln \left|x \right|$ from an approach similar to that of the squeeze theorem. Is it possible to ...
71 views

### The Riesz transform kernel satisfies Hörmander's condition

Define the kernel $$K_i(x) = \frac{x_i}{\lvert x\rvert^{d+1}}$$ as a function $\mathbb{R}^d \setminus \{ 0 \} \to \mathbb{C}$ for $d \geq 3$ (say). I want to prove that this kernel satisfies Hörmander'...
34 views

### Prove that orthonormalsystem is an orthonormalbasis [duplicate]

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
74 views

### Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
46 views

36 views

### Compact $K\subset A$ such that $\lambda(K) = \lambda(A) / 2$

Let $A\subset \mathbb{R}$ be a (Lebesgue) measurable set of finite measure. Using the fact that the function $f:\mathbb{R}\rightarrow \mathbb{R}$, $$f(x)=\lambda(A\cap [-x,x])$$ is continuous, we ...
Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
Consider the integral given by $$f(r)=\int_{0}^{\tanh(r)} \arccos\left(\frac{\sigma}{\sinh(r)\sqrt{1-\sigma^2}}\right)\cdot \frac{1}{\sqrt{\sigma^2+a^2}}d\sigma,$$ where $a>0$. I am wondering ...