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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13
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+100

Number of flips to get to a Set of Positive Lebesgue Measure

A consequence of Exercise 1.1.19 on page 13 of Stroock's "Probability Theory: An Analytic View" is that if a set $E\subset[0,1)$ has positive (Lebesgue) measure, then for almost every $x\in[0,1)$, a ...
2
votes
1answer
84 views
+50

If a joint cdf is increasing in each argument, then the pdf is strictly positive a.s.?

Let $F:\mathbb{R}^d \to [0,1]$ be an absolutely continuous joint cdf and let it be strictly increasing in each argument. Does it imply that its pdf $f$ is strictly positive a.s. (with respect to the ...
2
votes
0answers
76 views
+50

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
4
votes
0answers
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+50

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
12
votes
2answers
204 views
+50

Bernoulli's inequality and an unexpected limit

This question is inspired by What would happen to Bernoulli's inequality if $x<-1$?. Let $x_n=\min\{x\in{\bf R}:(1+x)^n\geq 1+nx\}$, where $n$ is natural and odd (my mistake in the first ...
2
votes
0answers
68 views
+50

Limit of a continuous function with a parameter

Let $f(x,\alpha)$ be continuous function on $S=(0,1]\times[0,1]$. Suppose that for every segment $[\alpha,\alpha+\Delta\alpha]\in[0,1]$ there exists $x_0=x_0(\Delta \alpha)$ s.t. for $0<x<x_0$ ...