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Lesson: Types of Discontinuities

In this lesson, we will learn how to differentiate between the three types of function discontinuity at a given point.

Sample Question Videos

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06:05

Worksheet • 24 Questions • 2 Videos

Q1:

Discuss the continuity of the function 𝑓 at π‘₯ = βˆ’ 2 , given

  • A The function is continuous at π‘₯ = βˆ’ 2 .
  • B The function is discontinuous at π‘₯ = βˆ’ 2 because l i m π‘₯ β†’ βˆ’ 2 𝑓 ( π‘₯ ) does not exist.
  • C The function is discontinuous at π‘₯ = βˆ’ 2 because 𝑓 ( βˆ’ 2 ) β‰  𝑓 ( π‘₯ ) l i m π‘₯ β†’ βˆ’ 2 .
  • D The function is discontinuous at π‘₯ = βˆ’ 2 because 𝑓 ( βˆ’ 2 ) is undefined.

Q2:

Given If possible or necessary, define 𝑓 ( βˆ’ 8 ) so that 𝑓 is continuous at π‘₯ = βˆ’ 8 .

  • AThe function cannot be made continuous at π‘₯ = βˆ’ 8 because l i m π‘₯ β†’ βˆ’ 8 + 𝑓 ( π‘₯ ) β‰  l i m π‘₯ β†’ βˆ’ 8 βˆ’ 𝑓 ( π‘₯ ) .
  • BThe function is already continuous at π‘₯ = βˆ’ 8 .
  • C 𝑓 ( βˆ’ 8 ) = 0 would make 𝑓 continuous at π‘₯ = βˆ’ 8 .
  • D 𝑓 ( βˆ’ 8 ) = 6 would make 𝑓 continuous at π‘₯ = βˆ’ 8 .

Q3:

Suppose What can be said of the continuity of 𝑓 at π‘₯ = βˆ’ 1 ?

  • A The function is continuous at π‘₯ = βˆ’ 1 .
  • B The function is discontinuous at π‘₯ = βˆ’ 1 because 𝑓 ( βˆ’ 1 ) β‰  𝑓 ( π‘₯ ) l i m π‘₯ β†’ βˆ’ 1 .
  • C The function is continuous on ℝ .
  • D The function is discontinuous at π‘₯ = βˆ’ 1 because 𝑓 ( βˆ’ 1 ) is undefined.
  • E The function is discontinuous at π‘₯ = βˆ’ 1 because l i m π‘₯ β†’ βˆ’ 1 𝑓 ( π‘₯ ) does not exist.

Q4:

Find the values of π‘Ž and 𝑏 that make the function 𝑓 continuous at π‘₯ = βˆ’ 1 and π‘₯ = βˆ’ 6 , given that

  • A π‘Ž = 1 2 5 , 𝑏 = 3 7 5
  • B π‘Ž = 1 2 5 , 𝑏 = βˆ’ 2 5 7 5
  • C π‘Ž = 3 7 5 , 𝑏 = 1 2 5
  • D π‘Ž = βˆ’ 6 1 7 , 𝑏 = βˆ’ 1 2 7
  • E π‘Ž = βˆ’ 1 2 7 , 𝑏 = βˆ’ 6 1 7

Q5:

Given 𝑓 ( π‘₯ ) = π‘₯ + π‘₯ βˆ’ 2 π‘₯ βˆ’ 1  , if possible or necessary, define 𝑓 ( 1 ) so that 𝑓 is continuous at π‘₯ = 1 .

  • A 𝑓 ( 1 ) = 3 makes 𝑓 continuous at π‘₯ = 1 .
  • BThe function is already continuous at π‘₯ = 1 .
  • CThe function cannot be made continuous at π‘₯ = 1 because 𝑓 ( 1 ) is undefined.
  • DNo value of 𝑓 ( 1 ) will make 𝑓 continuous because l i m  β†’  𝑓 ( π‘₯ ) does not exist.

Q6:

Given 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 4 π‘₯ + π‘₯ βˆ’ 2 0   , if possible or necessary, define 𝑓 ( 4 ) so that 𝑓 is continuous at π‘₯ = 4 .

  • A 𝑓 ( 4 ) = 1 6 3 makes 𝑓 continuous at π‘₯ = 4 .
  • BThe function is already continuous at π‘₯ = 4 .
  • CThe function cannot be made continuous at π‘₯ = 4 because 𝑓 ( 4 ) is undefined.
  • DNo value of 𝑓 ( 4 ) will make 𝑓 continuous because l i m  β†’ οŠͺ 𝑓 ( π‘₯ ) does not exist.

Q7:

Discuss the continuity of the function 𝑓 at π‘₯ = 5 given

  • A The function is discontinuous at π‘₯ = 5 because l i m π‘₯ β†’ 5 𝑓 ( π‘₯ ) does not exist.
  • B The function is discontinuous at π‘₯ = 5 because l i m π‘₯ β†’ 5 𝑓 ( π‘₯ ) β‰  𝑓 ( 5 ) .
  • C The function is discontinuous at π‘₯ = 5 because 𝑓 ( 5 ) is undefined.
  • D The function is continuous at π‘₯ = 5 .

Q8:

Find the value of π‘Ž that makes 𝑓 continuous at π‘₯ = 3 , given that

Q9:

Setting 𝑓 ( π‘Ž ) = 5 4 and 𝑓 ( π‘₯ ) = π‘₯ βˆ’ π‘Ž π‘₯ βˆ’ π‘Ž 6 6 3 3 when π‘₯ β‰  π‘Ž makes 𝑓 continuous at π‘₯ = π‘Ž . Determine π‘Ž .

  • A3
  • B 1 3
  • C 1 2
  • D2

Q10:

Consider the function

What is 𝑓 ( 0 ) ?

What is l i m π‘₯ β†’ 0 βˆ’ 𝑓 ( π‘₯ ) ?

What is l i m π‘₯ β†’ 0 + 𝑓 ( π‘₯ ) ?

What type of discontinuity does the function 𝑓 have at π‘₯ = 0 ?

  • AThe function 𝑓 has a removable discontinuity at π‘₯ = 0 .
  • BThe function 𝑓 does not have a discontinuity at π‘₯ = 0 .
  • CThe function 𝑓 has an essential discontinuity at π‘₯ = 0 .
  • DThe function 𝑓 has a jump discontinuity at π‘₯ = 0 .

Q11:

Consider the function

What is 𝑓 ( 0 ) ?

What is l i m π‘₯ β†’ 0 βˆ’ 𝑓 ( π‘₯ ) ?

  • AThe limit does not exist.
  • B1
  • C0
  • D βˆ’ ∞
  • E + ∞

What is l i m π‘₯ β†’ 0 + 𝑓 ( π‘₯ ) ?

  • A + ∞
  • B1
  • C0
  • D βˆ’ ∞
  • EThe limit does not exist.

What type of discontinuity does the function 𝑓 have at π‘₯ = 0 ?

  • AThe function 𝑓 has an essential discontinuity at π‘₯ = 0 .
  • BThe function 𝑓 has a jump discontinuity at π‘₯ = 0 .
  • CThe function 𝑓 has a removable discontinuity at π‘₯ = 0 .
  • DThe function 𝑓 does not have a discontinuity at π‘₯ = 0 .

Q12:

Determine the value of π‘Ž that makes 𝑓 continuous at π‘₯ = 0 , given

  • A βˆ’ 6 5 3
  • B βˆ’ 5 3
  • C 5 3
  • D 3 5 3
  • E βˆ’ 1 0

Q13:

Discuss the continuity of the function 𝑓 at π‘₯ = 0 , given

  • AThe function is discontinuous at π‘₯ = 0 because l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) β‰  𝑓 ( 0 ) .
  • BThe function is discontinuous at π‘₯ = 0 because 𝑓 ( 0 ) is undefined.
  • CThe function is continuous at π‘₯ = 0 .
  • DThe function is discontinuous at π‘₯ = 0 because l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist.

Q14:

The function is continuous at π‘₯ = 0 . Determine the possible values of π‘Ž .

  • A 3 , βˆ’ 3
  • B √ 3
  • C 2 , βˆ’ 2
  • D √ 3 , βˆ’ √ 3

Q15:

Determine the value of π‘Ž that makes the function 𝑓 continuous at π‘₯ = πœ‹ 4 , given

  • A 1 6
  • B 1 2
  • C2
  • D βˆ’ 1 6

Q16:

Find the value of π‘˜ that makes the function 𝑓 continuous at π‘₯ = πœ‹ 4 , given that

  • A βˆ’ 2 5
  • B βˆ’ 2
  • C βˆ’ 6
  • D βˆ’ 6 5

Q17:

Discuss the continuity of the function 𝑓 at π‘₯ = πœ‹ 2 , given

  • AThe function is continuous at π‘₯ = πœ‹ 2 .
  • BThe function is discontinuous on ℝ .
  • CThe function is discontinuous at π‘₯ = πœ‹ 2 because 𝑓 ο€» πœ‹ 2  is undefined.
  • DThe function is discontinuous at π‘₯ = πœ‹ 2 because l i m π‘₯ β†’ πœ‹ 2 𝑓 ( π‘₯ ) β‰  𝑓 ο€» πœ‹ 2  .
  • EThe function is discontinuous at π‘₯ = πœ‹ 2 because l i m π‘₯ β†’ πœ‹ 2 𝑓 ( π‘₯ ) does not exist.

Q18:

Determine the value of π‘Ž that makes the function 𝑓 continuous at π‘₯ = 0 , given that

  • A βˆ’ 2 1 8
  • B βˆ’ 4 5 8
  • C 3 3 8
  • D 2 7 8

Q19:

Find the value of π‘˜ which makes the function 𝑓 continuous at π‘₯ = 0 , given that

  • A 8 4 9
  • B 2 4 9
  • C 1 2
  • D 8 7
  • E2

Q20:

Discuss the continuity of the function 𝑓 at π‘₯ = πœ‹ 2 , given

  • AThe function is continuous at π‘₯ = πœ‹ 2 .
  • BThe function is discontinuous at π‘₯ = πœ‹ 2 because 𝑓 ο€» πœ‹ 2  is undefined.
  • CThe function is discontinuous at π‘₯ = πœ‹ 2 because l i m π‘₯ β†’ πœ‹ 2 𝑓 ( π‘₯ ) β‰  𝑓 ο€» πœ‹ 2  .
  • DThe function is discontinuous at π‘₯ = πœ‹ 2 because l i m π‘₯ β†’ πœ‹ 2 𝑓 ( π‘₯ ) does not exist.

Q21:

Let Find all values of that make continuous at .

  • A
  • B
  • C
  • D

Q22:

Let Find all values of π‘˜ that make 𝑓 continuous at π‘₯ = 0 .

  • A π‘˜ ∈ ℝ βˆ’ { 0 }
  • B0
  • C π‘˜ ∈ ℝ +
  • D π‘˜ ∈ ℝ

Q23:

Suppose What can be said of the continuity of 𝑓 at π‘₯ = 0 ?

  • A The function is continuous at π‘₯ = 0 .
  • B The function is discontinuous at π‘₯ = 0 because 𝑓 ( 0 ) β‰  𝑓 ( π‘₯ ) l i m π‘₯ β†’ 0 .
  • C The function is continuous on ℝ .
  • D The function is discontinuous at π‘₯ = 0 because 𝑓 ( 0 ) is undefined.
  • E The function is discontinuous at π‘₯ = 0 because l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist.

Q24:

Find the value of π‘˜ which makes the function 𝑓 continuous at π‘₯ = 4 , given

  • A 1 4
  • B βˆ’ 1
  • C βˆ’ 1 4
  • D βˆ’ 4
  • E4
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