# 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.

90,365 questions
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### Banach Fixed Point Theorem, System Has Solution

Using Banach's Fixed Point Theorem, show that the following system has at least one solution: $$x = 0.000001x^2 + 10\sin y + 1$$ $$y = 0.000001y^3 - 0.01\cos x - 1$$ Here is what I have tried: ...
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### Fourier transform of continuous and integrable function

$f\in L^1(R^n)$,and $f$ is continuous at 0,the Fourier transform of $f$ noted $\widehat{f}$ is non-negative,prove that$\widehat{f}\in L^1(R^n)$ I have no idea of how to use the condition of ...
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### Decide whether or not $g$ is differentiable at $0$

I need some help with an analysis problem. Define $g:[-1,1]\to \Bbb R$ by $g(x)=(-1)^k/k^2$ for $|x|\in(1/(k+1),1/k], k=1,2,...$ and $g(0)=0$. Decide whether or not $g$ is differentiable at $0$ and ...
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### A Proof about the Oscillation of a Function on Compact Interval

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r > 0$. Let $\text{osc}(f,c,r) = \sup |f(x)-f(y)|$, where the ...
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### Differentiability of partial function function

Let $E,F$ be Banach spaces. If $f\colon [0,1]\times E\to F$ is continuously differentiable, can we say something about the differentiability of $$g\colon E\to C^1([0,1],F),\quad y\mapsto f(\cdot,y)?$$ ...
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### Interior solution of a Variational Inequality in Hilbert Space

I'm trying to understand the proof of the following claim: Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$...
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### Limits of Rolle theorem

I would like to see a function $f:[a,b]\to\mathbb{R}$ that is differentiable in $(a,b)$ but it is not continuous at least at one of the interval boundary points $a$ or $b$. Can you show me one? This ...
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### Solving problems regarding $L(x)=\int_1^x\frac{dt}{t},\quad x>0$ and the inverse function $E(x)$

The problem Without using $L(x)=\ln(x)$ and given $$L(x)=\int_1^x\frac{dt}{t},\quad x>0$$ a) Prove that i) $L(xy)=L(x)+L(y),x,y>0$ ii) $L(1/x)=-L(x)$ b) Prove that $L(2)<1$ c) Prove ...
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### Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

IMPORTANT EDIT: this question has been moved to Mathoverflow It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a ...
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### Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Blockquote Theorem 1.25: Suppose $\phi \in L^1(\mathbb{R^n})$ and $\int_{\mathbb{R^n}} \phi =1$ . Also, let $\phi_{\epsilon}(x)=\frac{\phi(\frac{x}{\epsilon})}{\epsilon^n}$.Moreover , suppose ...
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### Differentiability implies continuity proof

Statement: If $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is differentiable at $\underline{x}$, it's continuous at $\underline{x}$. My proof: Since $f$ is differentiable at $\underline{x}$, all ...
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1answer
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### How to prove the finiteness of lower semi-continuous function?

Let $C$ be a convex set, and let $f$ be a lower semi-continuous convex function. In addition, we have $f$ is bounded above on every bounded subset of $\text{ri}~C$, where $\text{ri}~C$ is the relative ...
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### Quick way to compute $\int_{0}^{\infty} f(x) g(x+y)\, dx$ for $y \in [0,\infty)$

When we want to compute: $$h(y) = \int_0^y f(x) g(y-x)\, dx,$$ for $y \in [0,\infty)$ we can make use of a Fast Fourrier Transform to quickly compute these integrals. I have a similar problem where ...
1answer
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