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.
150,460 questions
0 votes
0 answers
21 views
Estimates in Hahn decomposition for measures of finite positive variation
(Hahn decomposition theorem) Let $\mu$ be a signed measure. Then one can find a partition $X = X_+ \cup X_-$ such that $\mu\downharpoonright_{X_+} \geq 0$ and $\mu\downharpoonright_{X_-} \leq 0$. A ...
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2 votes
2 answers
86 views
weaker condition that allows you to regroup a bounded number of terms in a series, without changing its nature or sum
While absolute convergence allows you to rearrange an infinite amount of terms as you please, generalizing both associativity and commutativity to infinite sums. Here we're only interested in grouping ...
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2 votes
1 answer
83 views
How are these integral behaviour approximations different
Q1. $$\int \frac{3x^{2} + 4x - 1}{(x^{2} + 1)^{2}\sqrt{x+1}}\, dx$$ $\textbf{A. }\frac{\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{B. }-\frac{2\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{C. }-\frac{x}{(x^{2}+1)\sqrt{x+...
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194 views
Question on the proof of Fundamental Theorem of Calculus
I'm reading this proof (i) of FTC in William R. Wade's Introduction to Analysis: I understand this proof, but when checking the definition of one-sided limit, I found that there's also a condition $a +...
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0 votes
2 answers
108 views
Definition of one sided limit
I am currently taking a real analysis class, and we've learned some notions from topology (nothing deep just key concepts). While I was proving a theorem, I came across something that really confused ...
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4 votes
0 answers
147 views
What is a simple necessary and sufficient condition for a function to be a derivative?
I am studying the characterization of derivatives in real analysis. I already know that if a function $f$ is a derivative of some function $F$, it must satisfy two conditions: 1.It must have the ...
2 votes
0 answers
109 views
About diagonal set
Let X be a compact Hausdorff space, and let $\Delta = \{ (x, x) \in X \times X \mid x \in X \}$ be the diagonal in $ X \times X$ . Consider the following two statements: 1.$X $is metrizable. 2.$ \...
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2 votes
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80 views
Convexity of the function $ f(x) = (1 - r^{1/x})^{x}$ on $[1,2]$ [closed]
How can I verify that the function $$ f(x) = \bigl(1 - r^{1/x}\bigr)^{x}, \qquad 0<r<1, $$ is convex on the interval $x \in [1,2]$? For example, one may take $r = 2/3$. Numerically the function ...
- 21
4 votes
1 answer
189 views
Proving that a sine function with countable discontinuities is Riemann integrable (without measure zero criterion)
How would I prove that the function $f\colon [0, 1]\to\mathbb{R}$ defined by \begin{equation} f(x) = \begin{cases} 1 & \text{if }x = \tfrac{1}{n} \text{ for some } n\in\mathbb{N}, \\ \sin (x) &...
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3 votes
2 answers
92 views
Stuck on proving the infinite partition of $\mathbb{N}$
I am currently self studying real analysis from the book Understanding Analysis, Stephen Abbott, 2nd edition. In page $11$, exercise $1.2.4$, the problem states: Produce an infinite collection of ...
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1 vote
1 answer
86 views
Downwards monotone convergence is unavailable for signed measures
For unsigned measure $\mu$ on a measurable space $(X, \mathcal B)$, we have: (Downwards monotone convergence) If ${E_1 \supset E_2 \supset \ldots}$ are ${{\mathcal B}}$-measurable, and ${\mu(E_n)<\...
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2 votes
1 answer
204 views
An efficient series for the Glaisher–Kinkelin constant?
From the following identity: $$ 3\ln A-\frac{1}{4}-\frac{1}{3}\ln 2=\int_0^1\frac{1}{\ln z}\left(\frac{1}{4}-\frac{1}{(1+z)^2}\right)dz$$ Is it viable to obtain a series expansion for $\ln(A)$? $A$ ...
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1 vote
0 answers
159 views
Is there a name for the limit $\lim_{\epsilon \to 0} \frac{f(a - \epsilon) + f(a + \epsilon)}{2}$?
If a real or complex function $f$ is undefined at a point $a$, but is sufficiently well-behaved near that point, we can find a sort of "average value" or "finite part" of $f$ at $a$...
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74 views
Why does the Taylor series of $f(x)=e^{-1/x^2}$ at $0$ converge everywhere but not represent the function? [duplicate]
I just encountered the function defined by $f(x)=e^{-1/x^2}$ for $x\neq0$ and $f(0)=0$. Every derivative of this function at $0$ is equal to $0$, so the Taylor series around $0$ is simply the zero ...
-1 votes
0 answers
45 views
Closed form for $\text{Li}_2(\varphi)$
Is it not trivial to derive a form for the Dilogarithm of the Golden Ratio ($\varphi$)? I simply plugged $\varphi$ into the known inverse Dilogarithm relation $\text{Li}_2(x)+\text{Li}_2\left(\frac1x\...
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