Articles of real analysis

Approximation by smooth function while preserving the zero set

Let $\mathbb{T}$ denote the unit circle. Given $f \in \mathcal{C}(\mathbb{T})$, can we approximate $f$ by smooth functions having the same zero set ? i.e. for $\varepsilon >0$, can we find $g \in \mathcal{C}^\infty(\mathbb{T})$ such that f(x) = 0 if and only if $g(x) = 0$; $\|f-g\|_\infty < \varepsilon$. Both tasks can easily be performed separately. […]

Continuity, Compactness and Graph

PMA, Rudin p.99 Exercise 6 Let $X,Y$ be metric spaces and $E$ be a compact subset of $X$. Define $f:E\rightarrow Y$ and $G=\{(x,f(x))\in X\times Y:x\in E\}$. Then prove that $f$ is continuous on $E$ iff $G$ is compact. I’m not sure hypotheses Rudin made are sufficient to prove this. How do I know what kind […]

Strictly convex function: how often can its second derivative be zero?

It’s a basic fact that a twice-differentiable function from $\mathbb{R}$ to $\mathbb{R}$ is strictly convex if its derivative is positive everywhere. The converse is not true: consider, e.g., $f(x) = x^4$, which is strictly converse, with $f ”(0)=0$. Is there a partial converse, however? Is it true, e.g., that a strictly convex twice-differentiable function from […]

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)= \operatorname{p.v.}\int_{\mathbb{R}}\frac{f(x-y)}{y} \, dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ space) ?

Integral equal $0$ for all $x$ implies $f=0$ a.e.

Let $f\in L^2[0,1]$. $$\int_0^x f(t)dt =0 \quad \forall\, x\in(0,1) \Longrightarrow f=0 \text{ a.e}$$ What is the easiest proof?

How calculate the indefinite integral $\int\frac{1}{x^3+x+1}dx$

How do I calculate the following indefinite integral? $$\int\frac{1}{x^3+x+1}dx$$ Approach: $x^3+x+1=(x-a)(x^2+ax+c)$ where $a:$ real solution of the equation $a^3+a+1=0$ $c:$ real solution of the equation $c^3-c^2+1=0$ Then $$\int\frac{1}{x^3+x+1}dx=\int\frac{1}{(x-a)(x^2+ax+c)}dx=\int\frac{A}{(x-a)}dx+\int\frac{Bx+C}{(x^2+ax+c)}dx$$

First order variation and total variation of a function/stochastic process

The notions of first-order variation and total variation of a function or a stochastic process are equated in this book. However, I found their definitions different from two other sources: In Wikipedia the total variation of a function $f$ from time $0$ to time $T$ is defined as $$ \sup_\Pi \sum_{i=0}^{n-1} | f(t_{i+1})-f(t_i) | . […]

Is $[0, 1) \times (0, 1)$ homeomorphic to $(0, 1) × (0, 1)$?

I know how to show $[0,1] \times [0,1]$ is not homeomorphic to $(0,1) \times (0,1)$ by a compactness argument. Is there such an argument that shows $[0,1) \times (0,1)$ is not homeomorphic to $(0,1) \times (0,1)$? If not, what is the best way to show that they’re not homeomorphic?

Help with epsilon-delta proof that 1/(x^2) is continuous at a point.

I’m trying to prove that $\lim_{x \to x_0} \frac{1}{ x^2 } = \frac{1}{ {x_0}^2 }$. I know this means that for all $\epsilon > 0$, I must show that there exists a $\delta > 0$ such that $\left | x – x_0 \right | < \delta \Rightarrow \left | \frac{1}{ x^2 } – \frac{1}{ {x_0}^2 […]

Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$

I would like to show that for $A(x) = \int_{0}^{x}\frac{1}{1+t^2}dt$, we have $A\left(\frac{2x}{1-x^2}\right)=2A(x)$, for all $|x|<1$. My idea is to start with either $2\int_0^x\frac{1}{1+t^2}dt$ or $\int_0^{2x/(1-x^2)}\frac{1}{1+t^2}dt$, and try to transform one into the other by change of variables. (It would make more sense for the moment if we did not do any trigonometric substitutions, since […]