Q: Let $$f(x) = \begin{cases} x \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}. Is$$ $$f$$ continuous at $$x = 0$$?

Yes, because $$\lim\limits_{x \to 0} f(x) = f(0).$$

Question 1

Which of the following statements about the set of rational numbers $$\mathbb{Q} is $$TRUE?

Accepted Answer

Between any two rational numbers, there exists another rational number.

Question 2

Let $$A = \{ x \in \mathbb{R} : x \u003c 2 \}. $$Which of the following is the supremum (least upper bound) of $$A$$?

Accepted Answer

$$2$$

Question 3

Consider the sequence $$a_n = (-1)^n. $$Which of the following best describes its behavior as $$n 	o \infty$$?

Accepted Answer

It diverges because it oscillates between $$1$$ and $$-1.$$

Question 4

Which of the following intervals is both open and closed in $$\mathbb{R}$$?

Accepted Answer

$$(−\infty, \infty)$$

Question 5

Let $$f(x) = \frac{x^2$ - 1}{x - 1}. $$Which of the following statements is TRUE?

Accepted Answer

The function has a removable discontinuity at $$x = 1.$$

Question 6

Suppose $$f : \mathbb{R} 	o \mathbb{R} is $$continuous and $$g : \mathbb{R} 	o \mathbb{R} is $$continuous. Which of the following is NOT necessarily continuous?

Accepted Answer

$$\frac{f}{g}$$, where $$g(x) = 0$$ for some $$x \in \mathbb{R}$$

Question 7

Let $$f(x) = \begin{cases} x \sin\left(\frac{1}{x}ight), & x 
eq 0 \ 0, & x = 0 \end{cases}. Is$$ $$f$$ continuous at $$x = 0$$?

Accepted Answer

Yes, because $$\lim\limits_{x 	o 0} f(x) = f(0).$$

Question 8

Which of the following best describes the set $$\mathbb{Q} ($$the rationals) as a subset of $$\mathbb{R}$$?

Accepted Answer

$$\mathbb{Q}$$ has no interior points but every point is a limit point.

Question 9

Let $$f(x) = |x|/x. $$What type of discontinuity does $$f$$ have at $$x = 0$$?

Accepted Answer

Jump discontinuity

Question 10

Suppose $$A = \{1/n : n \in \mathbb{N}\}. $$What is $$\inf A$$ and $$\sup A$$?

Accepted Answer

$$\inf A = 0$$, $$\sup A = 1$$

Question 11

Let $$f(x) = \begin{cases} 1, & x \in \mathbb{Q} \ 0, & x 
otin \mathbb{Q} \end{cases}. At $$which points is $$f$$ continuous?

Accepted Answer

At no $$x \in \mathbb{R}$$, except possibly at $$x = 0$$

Question 12

Which of the following is the correct definition of continuity of a function $$f at a $$point $$x_0$$?

Accepted Answer

For all $$\varepsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$|x - x_0$| \u003c \delta$$ and $$x \in D(f)$$, then $$|f(x) - f(x_0$)| \u003c \varepsilon.$$

Question 13

Let $$f(x) = \frac{1}{x}. $$Which of the following is TRUE?

Accepted Answer

$$f is $$continuous on $$\mathbb{R} \setminus \{0\}.$$

Question 14

Suppose $$f(x) is $$defined on $$\mathbb{R}$$ and $$\lim\limits_{x 	o x_0$} f(x) = L. $$Which of the following must be true for $$f to be $$continuous at $$x_0$$?

Accepted Answer

$$f(x_0$) = L$$

Question 15

Let $$f(x) = \begin{cases} 1/x, & x 
eq 0 \ 0, & x = 0 \end{cases}. $$What type of discontinuity does $$f$$ have at $$x = 0$$?

Accepted Answer

Infinite discontinuity

Question 16

Which of the following is a correct set builder notation for the interval $$[−5, \infty)$$?

Accepted Answer

$$\{ x \in \mathbb{R} : x \geq -5 \}$$

Question 17

Let $$f(x) = 	an(x). $$Which of the following statements is TRUE?

Accepted Answer

$$f is $$continuous except at $$x = \frac{\pi}{2} + n\pi$$, $$n \in \mathbb{Z}$$, where it has infinite discontinuities.

Question 18

Let $$f(x) = \begin{cases} x, & x \in \mathbb{Q} \ 0, & x 
otin \mathbb{Q} \end{cases}. At $$which point is $$f$$ continuous?

Accepted Answer

At $$x = 0$$ only.

Question 19

Suppose $$f(x) is $$continuous on $$\mathbb{R}$$ and $$f(x) = 0$$ for all $$x \in \mathbb{R}. $$Which of the following is TRUE?

Accepted Answer

The function is continuous everywhere.

Real Numbers, Limits, and Continuity: Foundations for Calculus

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