Question 1

Which of the following statements is true about the continuity of a function at a point?

A function is continuous at a point if its limit at that point exists.
A function is continuous at a point if its left-hand limit and right-hand limit at that point are equal.
A function is continuous at a point if its derivative exists at that point.
A function is continuous at a point if its graph has no breaks or jumps at that point.

Answer

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Question 2

What is the epsilon-delta definition of continuity?

For every epsilon > 0, there exists a delta > 0 such that if 0 < |x - c| < delta, then |f(x) - f(c)| < epsilon.
For every epsilon > 0, there exists a delta > 0 such that if 0 < |x - c| < delta, then |f(x) - L| < epsilon, where L is the limit of f(x) at c.
For every epsilon > 0, there exists a delta > 0 such that if 0 < |x - c| < delta, then |f(x) - f(c)| > epsilon.
For every epsilon > 0, there exists a delta > 0 such that if 0 < |x - c| < delta, then |f(x) - L| > epsilon, where L is the limit of f(x) at c.

Answer

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Question 3

Which of the following functions is continuous at x = 0?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

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Question 4

Which of the following functions is discontinuous at x = 0?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

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Question 5

What is the type of discontinuity of the function f(x) = 1/x at x = 0?

Removable discontinuity
Jump discontinuity
Infinite discontinuity
Oscillating discontinuity

Answer

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Question 6

Which of the following functions has a jump discontinuity at x = 1?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

Answer

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Question 7

Which of the following functions has an oscillating discontinuity at x = 0?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(1/x)

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Question 8

Which of the following statements is true about the continuity of a function on an interval?

A function is continuous on an interval if it is continuous at every point in the interval.
A function is continuous on an interval if it is continuous at every rational number in the interval.
A function is continuous on an interval if it is continuous at every irrational number in the interval.
A function is continuous on an interval if it is continuous at every point except for a finite number of points.

Answer

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Question 9

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

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

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Question 10

Which of the following functions is discontinuous on the interval [0, 1]?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

Answer

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Question 11

Which of the following statements is true about the continuity of a function on a closed interval?

A function is continuous on a closed interval if it is continuous at every point in the interval, including the endpoints.
A function is continuous on a closed interval if it is continuous at every point in the interval, except for the endpoints.
A function is continuous on a closed interval if it is continuous at every rational number in the interval, including the endpoints.
A function is continuous on a closed interval if it is continuous at every irrational number in the interval, including the endpoints.

Answer

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Question 12

Which of the following functions is continuous on the closed interval [0, 1]?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

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Question 13

Which of the following functions is discontinuous on the closed interval [0, 1]?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

Answer

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Question 14

Which of the following statements is true about the continuity of a function on an open interval?

A function is continuous on an open interval if it is continuous at every point in the interval.
A function is continuous on an open interval if it is continuous at every rational number in the interval.
A function is continuous on an open interval if it is continuous at every irrational number in the interval.
A function is continuous on an open interval if it is continuous at every point except for a finite number of points.

Answer

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Question 15

Which of the following functions is continuous on the open interval (0, 1)?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

Answer

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Question 16

Which of the following functions is discontinuous on the open interval (0, 1)?

f(x) = x^2
f(x) = 1/x
f(x) = |x|
f(x) = sin(x)

Answer

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Description: This quiz is designed to assess your understanding of the concept of continuity in real analysis. It covers various aspects of continuity, including limits, epsilon-delta definition, and types of discontinuities.
Number of Questions: 16
Created by: Aliensbrain Bot
Tags: real analysis continuity limits epsilon-delta definition types of discontinuities

Continuity

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