Heine (sequential) criterion for limits and continuity

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Problem 1.10.12 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.10.2 "Problems"

Formulate the definition of a function continuous at a point, using the Heine definition of the limit of a function (i.e., in terms of sequences).

Problem 1.10.145 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.10.2 "Problems"

(1)(2)

(1)

Let $\lim_{n \rightarrow \infty } x_{n} = \infty$ . Prove that $\lim_{n \rightarrow \infty }\left(1+\frac{1}{x_{n}}\right)^{x_{n}} = e$ .

(2)

Let $\lim_{n \rightarrow \infty } x_{n} = 0, x_{n} \neq 0$ . Prove that $\lim_{n \rightarrow \infty }\left(1+x_{n}\right)^{1 / x_{n}} = e$ .

Problem 1.9.7 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.9.2 "Problems"

(1)(2)

Prove that $\lim_{x \rightarrow 0} f(x)$ does not exist, if:

(1)

$f(x) = \operatorname {arcctg}(1 / x)$ ;

(2)

$f(x) = \operatorname {sign} \sin (1 / x)$ .

Problem 1.9.8 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.9.2 "Problems"

(1)(2)

Prove that $\lim_{x \rightarrow +\infty } f(x)$ does not exist, if:

(1)

$f(x) = \cos x$ ;

(2)

$f(x) = x-\mathbb {E}\left[x\right]$ .

Problem 1.9.10 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.9.2 "Problems"

Prove that if, for every sequence $\left\{ x_{n}\right\}$ converging to the point $x_{0}$ , the sequence $\left\{ f\left(x_{n}\right)\right\}$ converges, then $\lim_{x \rightarrow x_{0}} f(x)$ exists.

Problem 1.9.11 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.9.2 "Problems"

Prove that if from every sequence $\left\{ x_{n}\right\}$ converging to the point $x_{0}$ one can extract a subsequence $\left\{ x_{n_{k}}\right\}$ for which $\lim_{k \rightarrow \infty } f\left(x_{n_{k}}\right) = a$ , then $\lim_{x \rightarrow x_{0}} f(x) = a$ .

Problem 3.18.168 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03b]More problems in Subsection 3.18.2 "Problems"

Prove the stated assertion ( $X, Y, Z$ are metric spaces). The definitions of continuity of a mapping $f: X \rightarrow Y$ given above in terms of sequences and in terms of neighborhoods are equivalent.

Problem 2.21 from Athreya K.B., Lahiri S.N. "Measure Theory and Probability Theory" [AtLa06]More problems in Chapter 2 "Integration"

(a)(b)

Let $g: \mathbb {R} \rightarrow \mathbb {R}$ .

(a)

Recall that $\lim_{t \rightarrow \infty } g(t) = L$ for some $L$ in $\mathbb {R}$ if for every $\epsilon > 0$ , there exists a $T_{\epsilon } < \infty$ such that $t \geq T_{\epsilon } \Rightarrow \left|g(t)-L\right| < \epsilon$ . Show that $\lim_{t \rightarrow \infty } g(t) = L$ for some $L$ in $\mathbb {R}$ iff $\lim_{n \rightarrow \infty } g\left(t_{n}\right) = L$ for every sequence $\left\{ t_{n}\right\}_{n \geq 1}$ with $\lim_{n \rightarrow \infty } t_{n} = \infty$ .

(b)

Formulate and prove a similar result when $\lim_{t \rightarrow a} g(t) = L$ for some $a, L \in \mathbb {R}$ .

References

[ККЧ+03a] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.

[ККЧ+03b] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[AtLa06] Athreya K.B., Lahiri S.N. Measure Theory and Probability Theory. Springer, 2006.