# 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.

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### What is the realcompactification of the real line?

I've studied the definition of the Stone-Cech compactification by Munkres Topology. I have realized that we can't write the Stone-Cech compactification of the real line explicitly, we are just able to ...
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### Equivalence of measurability? What is the advantage of each of these?

So i am reading Royden and he introduces measurability via Cartheodory. $E \subset \mathbb{R}$ is measurable $\iff$ for any set $A \subset \mathbb{R}$ $m^*(A) = m^*(E\cap A) + m^*(E\cap A^c)$. ...
1answer
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### Counterexample to Uniform Convergence of Convex Functions

It is stated here that if a sequence $f_n : [a,b] \to \mathbb{R}$ of convex continuous functions converges pointwise to a continuous function $f$, then the convergence is in fact uniform. The ...
2answers
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3answers
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### The only subspace of $\mathbb{R}$ that is homeomorphic to $\mathbb{R}$ and complete (with the restricted metric) is $\mathbb{R}$

The only subspace of $\mathbb{R}$ that is homeomorphic to $\mathbb{R}$ and complete (with the restricted metric) is $\mathbb{R}$. My work- Let $A$ be a subspace of $\mathbb{R}$ such that $A$ is ...
1answer
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### If $L^1$ average of $f$ is smaller than $t^2$ then $f=0$ a.e.

Suppose $f\in L^1(\mathbb R)$ and there exits $\delta>0$ such that $$\|f(\cdot + t)-f\|_{L^1}\leq |t|^2$$ for all $|t|\leq \delta$, where $f(\cdot + t)$ is the function $x\mapsto f(x+t)$. Show ...
4answers
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### Prove that if $f=f'$ then $f$ is monotone.

Suppose I do not know that $e^x$ solves the equation $$f'(x)=f(x),\;\;\;x\in\mathbb{R}.$$ I am just given this equation and want to see if $f$ is increasing. Is there a way to prove that $f$ is ...
2answers
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### Mean field games and approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf. I have a question about a step in theorem 3.8 on page 17. Let ...
1answer
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### Topology of point wise convergence

I encountered this topology in the exercise below. I was wondering if there is any connection to algebraic geometry? This looks like algebraic geometry but instead of points we are dealing with ...