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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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Differentiability of $f(x)= 2-\mathrm{e}^{-x}$ when $x\geq 0$, $\mathrm{e}^{-x}$ otherwise

I want to check for any $x_0$ in its domain, whether this function is differentiable or not. $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto \begin{cases} 2-\mathrm{e}^{-x} & ;\ x\geq 0 \\ \mathrm{e}^{-x} & ; x < 0 …
fear.xD
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An example of a bijective function with an infinite number of discontinuity

I am trying to do the following exercice : 1) Show that $f:\mathbb{R}\rightarrow[0,+∞)$,bijective, has an infinite number of discontinuity. 2) An example ? My work: 1) I have succeeded by contradiction and use the fact that f continuous and…
user119228
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An Integrable function which is not essentially bounded on any subinterval of $[0,1]$

I am looking for an Integrable function $f : [0,1] \to \mathbb{R}$ which is not essentially bounded on any subinterval of $[0,1]$. Thank you in advance.
the8thone
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Find a differentiable $f$ such that $\mathrm{Zeros}(f)=\mathbb{any\; closed \;set}$

Let $B\subset \mathbb{R}^2$ be a closed set. How to prove that there is a differentiable function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $$Z(f)=B$$ where $$Z(f)=\{x\in\mathbb{R}^2:f(x)=0\}$$ Any hints would be appreciated.
felipeuni
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How to simplify $\tan{\arcsin{\frac{y}{R}}/2}$?

I have verified with Mathematica that, for $R>0, y \in \mathbb{R}$: $$ \tan{\frac{\arcsin{\frac{y}{R}}}{2}} = \frac{R - \sqrt{R^2 -y ^2}}{y}$$ using Assuming[Element[y, Reals] && R > 0, FullSimplify[TrigToExp[Tan[ArcSin[y/R]/2]]]] How can I…
José D.
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Show that a continuous function with a certain integral property must be f(x)=x.

I am studying for the quals in January and am working through the following problem: Let $f:[0,1] \to [0,1]$ be a continuous function with the property that $\displaystyle\int_0^1 f(x)x^n \, \textrm{d}x = \dfrac{1}{n+2}$. Show that $f(x)\equiv…
Tyler Clark
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Essential conditions for Stone-Weierstrass theorem

Recently, i've been reviewing analysis. And i found this theorem in my text and that in wikipedia differ. Indeed, wikipedia one is strictly stronger. ======= Rudin - PMA p.162 Let $X$ be a compact Hausdorff space. Let $(C(X,\mathbb{R}),||•||)$ be…
Jj-
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real analysis derivatives and continuity

Suppose $g \colon[a,b] \rightarrow \mathbb{R}$ is continuous on $[a,b]$ and integrable with $g(a)=g(b)=0$ and that $g'' \colon (a,b)\rightarrow \mathbb{R}$ is continuous on $(a,b)$ and bounded by $K$. Prove that $|g(x)| \leq K|x-a||x-b|$ for all $x…
user115810
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How to show $\gamma$ aka Euler's constant is convergent?

I'm trying to show different convergents, and this is the first one i'm having problems with. $\dfrac11 + \dfrac 12 + \ldots + \dfrac 1n - \log n \rightarrow \gamma$ Like it's the definition of euler's constant, but how to show that this…
Krzysztof Lewko
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My (simpler) proof of the divergence of a harmonic series.

Let $H=1+\frac{1}{2}+\frac{1}{3}+\cdots$ . Proving that it diverges this is what I did. I supposed that the series converges to $H$, …
Module
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find the limit of (f(x) - f(-x)) /x when x goes to zero

Please some one help me to do the following problem. If $f : \mathbb R\to\mathbb R$ is differentiable at $0$ and $f'(0) =1$, find $\lim\limits_{x\to 0} \frac{f(x) - f(-x)}x$.
Chans
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Show inequality

I want to how nicely define the $f(x)$ for this type of question to prove the inequality use the mean value theorem $$e^x \ge 1+x ,\ x \in \mathbb{R}$$ How to choose $f(x)$ to show that inequality and do $$f(b) - f(a) = f'(c)(b-a),\…
Kimchi
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Let $\,f \colon \Bbb R \to \Bbb R$ be a continuous function such that $|f(x)-f(y)|\ge \frac12 |x-y|$

I am stuck on the following problem that says: Let $\,f \colon \Bbb R \to \Bbb R$ be a continuous function such that $\,|f(x)-f(y)|\ge \frac12 |x-y|, \forall x,y \in \Bbb R$ . Then which of the following options is correct? $f$ is both one-to-one …
learner
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A "simple" functional equation

In a literary work "functional equation", I found a functional equation may be difficult to me, that is, provided that a real differential function on the real line I think maybe it needs first to prove the map is surjection or injection? I know…
David Chan
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Bounded variation functions have jump-type discontinuities

I read on the Wikipedia page that bounded variation (BV) functions have only jump-type discontinuities. Why is that? Suppose at some $a\in\mathbb{R}$, the limit $\lim_{x\rightarrow a^+}f(x)$ doesn't exist (or is infinite). Why would such a function…
JJ Beck
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