Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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3
votes
1answer
124 views
+100

Conditions on $f:[a,b]\to \mathbb{R}$ to ensure a finite number of crossings

Suppose $f:[a,b]\to \mathbb{R}$ is differentiable except at a finite number of points. Can you give me some examples of additional hypotheses that would guarantee that $$\text{For any $c\in ...
7
votes
1answer
602 views
+200

Existence of a Strictly Increasing, Continuous Function whose Derivative is 0 a.e. on $\mathbb{R}$

This proof is almost done except for the step of showing that the function's derivative is $0$ a.e. Let $I = \{[p_n, q_n]\}$ denote the set of all closed intervals in $\mathbb{R}$ with rational ...
4
votes
2answers
118 views
+50

Convergence of the series $\sum_{n=0}^\infty \frac{1}{n+1}\sin\bigr(\frac{p\pi u_n}{q}\bigl)$

Let $(u_n)_{n\in \mathbb{N}}$ defined by : $u_0=1, u_1=1$ and for all integer $u_{n+1}=3u_n-u_{n-1}$ Study the convergence of $$\displaystyle\sum_{n=0}^\infty ...
9
votes
0answers
206 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
8
votes
0answers
221 views
+400

$\int_0^1 [ \frac{1}{x(x-1)} (2\mathrm{Li}_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
0
votes
1answer
48 views
+50

$C^\omega$ notation for real analytic functions

I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used ...
3
votes
1answer
152 views
+50

Cantor set exercise

This is an exercise from Abbott's real analysis book. It's exercise 3.4.4.(b) on page 93. I couldn't find a definition of ''dimension'' in the book. The only thing I could find is something on page ...