Q: Which of the following statements best describes the formal definition of continuity at a point $$x_0$$ for a function $$f: X \to \mathbb{R}$$?

$$f is $$continuous at $$x_0 if$$ $$\lim_{x \to x_0$; x \in X} f(x) = f(x_0$).$$

Question 1

Which of the following statements best describes the formal definition of continuity at a point $$x_0$$ for a function $$f: X 	o \mathbb{R}$$?

Accepted Answer

$$f is $$continuous at $$x_0 if$$ $$\lim_{x 	o x_0$; x \in X} f(x) = f(x_0$).$$

Question 2

Which of the following functions is continuous everywhere on $$\mathbb{R}$$?

Accepted Answer

$$f(x) = x$$

Question 3

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

Accepted Answer

Removable discontinuity

Question 4

Let $$f(x) = x^2$ on $$[0, \infty)$$. Which of the following best describes its monotonicity?

Accepted Answer

$$f is $$strictly increasing on $$[0, \infty).$$

Question 5

Which of the following is a correct equivalent formulation of continuity at $$x_0$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|x - x_0$| \u003c \delta$$ implies $$|f(x) - f(x_0$)| \u003c \epsilon.$$

Question 6

Let $$f(x) = \frac{1}{x} on$$ $$(0, 1). $$Which of the following statements is true?

Accepted Answer

$$f is $$continuous but not bounded on $$(0, 1).$$

Question 7

Suppose $$f is $$continuous on $$[a, b]. $$Which of the following is guaranteed by the maximum principle?

Accepted Answer

$$f$$ attains both a maximum and a minimum on $$[a, b].$$

Question 8

Let $$f: [0, 2] 	o \mathbb{R} be $$continuous with $$f(0) = 1$$ and $$f(2) = 5. $$Which of the following statements is a direct consequence of the Intermediate Value Theorem?

Accepted Answer

There exists $$c \in [0, 2]$$ such that $$f(c) = 3.$$

Question 9

Which of the following functions is uniformly continuous on $$[0, 1]$$?

Accepted Answer

$$f(x) = x^2$

Question 10

Let $$f be $$continuous and strictly increasing on $$[a, b]. $$Which of the following is true?

Accepted Answer

$$f^{-1}$$ exists, is continuous, and strictly increasing on $$[f(a), f(b)].$$

Question 11

Which of the following best describes the difference between continuity and uniform continuity?

Accepted Answer

Uniform continuity requires the $$\delta in $$the $$\epsilon$$-$$\delta$$ definition to depend only on $$\epsilon$$, not on the point $$x_0.$$

Question 12

Suppose $$f(x) = x^2$ on $$\mathbb{R}$$. Is f$ uniformly continuous on $$\mathbb{R}$$?

Accepted Answer

No, because the difference $$|f(y_n) - f(x_n)|$$ can be bounded away from zero for equivalent sequences.

Question 13

Let $$f(x) = \$$max(x^3$$ + 4x^2$ + x + 5, x^4$ - x^3$)$$ and $$g(x) = x^2$ - 4. On $$which set is $$\frac{f(x)}{g(x)}$$ continuous?

Accepted Answer

$$\mathbb{R} \setminus \{-2, 2\}$$

Question 14

Which of the following is a correct definition of the right-hand limit of $$f at$$ $$x_0$$?

Accepted Answer

$$\lim_{x 	o x_0$; x \in X \cap (x_0$, \infty)} f(x)$$

Question 15

Let $$f(x) = 	ext{sgn}(x). $$What are the left and right limits at $$x = 0$$?

Accepted Answer

Left limit is $$-1$$, right limit is $$1.$$

Question 16

Which of the following is a consequence of the Intermediate Value Theorem for a continuous function $$f: [a, b] 	o \mathbb{R}$$?

Accepted Answer

For any $$y$$ between $$f(a)$$ and $$f(b)$$, there exists $$c \in [a, b]$$ such that $$f(c) = y.$$

Question 17

Let $$f(x) = \frac{1}{x} on$$ $$(0, \infty). $$What is $$\lim_{x 	o +\infty} f(x)$$?

Accepted Answer

$$0$$

Continuity and Related Concepts in Real Analysis

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