7. Continuity

Sequential Characterization — Quiz

Test your understanding of sequential characterization with 5 practice questions.

Practice Questions

Question 1

Which statement best describes the sequential characterization of continuity of $f$ at $a$?

Question 2

If for every sequence $x_n\to a$ in the domain of $f$ we have $f(x_n)\to f(a)$, what can we conclude?

Question 3

Let $f(x)=x^2$ and let $x_n=1+\frac{1}{n}$. What is $\lim_{n\to\infty} f(x_n)$?

Question 4

Suppose $f(0)=1$ and $f(x)=0$ for all $x\neq0$. Let $x_n=\frac{1}{n}$. What does the sequential characterization show about $f$ at $0$?

Question 5

If there exists a sequence $x_n\to a$ such that $f(x_n)\not\to f(a)$, which conclusion is valid?