CLASSIFYING TYPES OF DISCONTINIUITY PRACTICE PROBLEMS

Discuss the continuity. If a discontinuity exists, then describe the type of discontinuity and its physical feature on a graph.

Problem 1 :

Solution

Problem 2 :

Solution

Problem 3 :

Solution

Problem 4 :

Solution

Problem 5 :

Solution

Problem 6 :

Solution

1)  Removable discontinuity at x = 3 or hole is at x = 3.

Non removable discontinuity is at x = 1.

2)  Non removable discontinuity is at x = 3 or jump discontinuity is at x = 3.

3)  Non removable discontinuity is at x = 3 or jump discontinuity is at x = 3.

4)  Removable discontinuity is at x = 0 or hole is at x = 0.

5)  Removable discontinuity is at x = 1 or hole is at x = 1.

6)  The function is continuous.

For each function identify the type of discontinuity and where it is located.

Problem 1 :

Solution

Problem 2 :

Solution

Problem 3 :

f(x) = x3 - 4x

Solution

Problem 4 :

Solution

Problem 5 :

f(x) = sec 2x for 0 ≤ x ≤ 2π

Solution

Problem 6 :

Solution

Problem 7 :

Solution

Problem 8 :

Solution

Problem 9 :

Solution

Problem 10 :

f(x) = csc (x/2) for 0 ≤ x ≤ 2π

Solution

Problem 11 :

The graph of the function 𝑓(𝑥) is shown to the right: Which of the following statements is true about 𝑓?

I. 𝑓 is undefined at 𝑥 = 1.

II. 𝑓 is defined but not continuous at 𝑥 = 2.

III. 𝑓 is defined and continuous at 𝑥 = 3.

(A) Only I     (B) Only II     (C) I and II

(D) I and III      (E) None of the statements are true.

Solution

Problem 12 :

The function f(x) has removable discontinuity at

(A) x = -2 only     (B) x = 0 only    (C) x = 1 only

(D) x = -2 and x = 0 only

(E) f(x) has no removable discontinuities

Solution

Problem 13 :

On what intervals is f(x) continuous ?

a) [-3, -2] U [-2, 0] U [0, 2.5]      b) [-3, -2] U (-2, 0] U [0, 2.5]

c) [-3, -2] U (-2, 0] U (0, 2.5]  d) [-3, -2] U [-2, 0] U (0, 2.5]

Solution

Problem 14 :

The function has jump discontinuity is at ?

Solution

1)  non removable discontinuity at x = -1. Vertical asymptote is at x = -1.

2)  Removable discontinuity at x, non removable discontinuity at x = -3.

3)  the function is continuous for all real values .

4)  removable discontinuity or hole at x = -3

5)  non removable discontinuities at x = π/4, 3π/4, 5π/4 and 7π/4.

6)  Removable discontinuity at x = -2, Non removable discontinuity or hole is at x = 4.

7)  Removable discontinuity at x = 5, Non removable discontinuity or hole is at x = 2.

8)  there is non removable discontinuity is at x = 5/2.

9)  the function is continuous for all real values.

10) Vertical asymptotes are at x = 0, 2π

11)  none of the statement are true, E

12)  option E is correct.

13)  Between 0 to 2.5, it is continuous. So, option c is correct.

14)  At x = -2 and at x = 0, the function has jump discontinuities.

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