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# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Representation of continuous, monotone, concave functions

Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying: $f(0)=0$ $f$ is monotonically increasing $f$ is concave My intuition is that $f$ should admit ...
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### Proof that the infimum of a set is incorrect [closed]

So we will say that we have a set S⊆R so that Inf(S)=p then n∈S ∃ε>0∈R s.t. n>ε+p <—-> inf(S)≠p. (Is this a good proof, please comment if there are any mistakes or improvements to make, I’...
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### Uniformly closed ideals of smooth/real analytic functions

Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
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### If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?

The question is as in the title. Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
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### Proof of the inequality $\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x}$ when $x,y \in (0,1]$

I am trying to prove the following inequality: $$\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x} \quad \forall x,y \in (0,1]$$ This inequality appears in the paper "...
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