Worksheet: Continuity at a Point

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In this worksheet, we will practice checking the continuity of a function at a given point.

Q1:

Discuss the continuity of the function 𝑓 at 𝑥 = − 2 , given

  • A The function is discontinuous at 𝑥 = − 2 because 𝑓 ( − 2 ) is undefined.
  • B The function is discontinuous at 𝑥 = − 2 because l i m 𝑥 → − 2 𝑓 ( 𝑥 ) does not exist.
  • C The function is discontinuous at 𝑥 = − 2 because 𝑓 ( − 2 ) ≠ 𝑓 ( 𝑥 ) l i m 𝑥 → − 2 .
  • D The function is continuous at 𝑥 = − 2 .

Q2:

Given If possible or necessary, define 𝑓 ( − 8 ) so that 𝑓 is continuous at 𝑥 = − 8 .

  • A 𝑓 ( − 8 ) = 6 would make 𝑓 continuous at 𝑥 = − 8 .
  • BThe function is already continuous at 𝑥 = − 8 .
  • C 𝑓 ( − 8 ) = 0 would make 𝑓 continuous at 𝑥 = − 8 .
  • DThe function cannot be made continuous at 𝑥 = − 8 because l i m 𝑥 → − 8 + 𝑓 ( 𝑥 ) ≠ l i m 𝑥 → − 8 − 𝑓 ( 𝑥 ) .

Q3:

Suppose What can be said of the continuity of 𝑓 at 𝑥 = − 1 ?

  • A The function is discontinuous at 𝑥 = − 1 because l i m 𝑥 → − 1 𝑓 ( 𝑥 ) does not exist.
  • B The function is continuous on ℝ .
  • C The function is discontinuous at 𝑥 = − 1 because 𝑓 ( − 1 ) is undefined.
  • D The function is continuous at 𝑥 = − 1 .
  • E The function is discontinuous at 𝑥 = − 1 because 𝑓 ( − 1 ) ≠ 𝑓 ( 𝑥 ) l i m 𝑥 → − 1 .

Q4:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥 = − 1 and 𝑥 = − 6 , given that

  • A 𝑎 = − 1 2 7 , 𝑏 = − 6 1 7
  • B 𝑎 = 3 7 5 , 𝑏 = 1 2 5
  • C 𝑎 = − 6 1 7 , 𝑏 = − 1 2 7
  • D 𝑎 = 1 2 5 , 𝑏 = 3 7 5
  • E 𝑎 = 1 2 5 , 𝑏 = − 2 5 7 5

Q5:

Given 𝑓 ( 𝑥 ) = 𝑥 + 𝑥 − 2 𝑥 − 1 , if possible or necessary, define 𝑓 ( 1 ) so that 𝑓 is continuous at 𝑥 = 1 .

  • ANo value of 𝑓 ( 1 ) will make 𝑓 continuous because l i m → 𝑓 ( 𝑥 ) does not exist.
  • BThe function is already continuous at 𝑥 = 1 .
  • CThe function cannot be made continuous at 𝑥 = 1 because 𝑓 ( 1 ) is undefined.
  • D 𝑓 ( 1 ) = 3 makes 𝑓 continuous at 𝑥 = 1 .

Q6:

Given 𝑓 ( 𝑥 ) = 𝑥 − 6 4 𝑥 + 𝑥 − 2 0 , if possible or necessary, define 𝑓 ( 4 ) so that 𝑓 is continuous at 𝑥 = 4 .

  • ANo value of 𝑓 ( 4 ) will make 𝑓 continuous because l i m → 𝑓 ( 𝑥 ) does not exist.
  • BThe function is already continuous at 𝑥 = 4 .
  • CThe function cannot be made continuous at 𝑥 = 4 because 𝑓 ( 4 ) is undefined.
  • D 𝑓 ( 4 ) = 1 6 3 makes 𝑓 continuous at 𝑥 = 4 .

Q7:

Discuss the continuity of the function 𝑓 at 𝑥 = 5 given

  • A The function is continuous at 𝑥 = 5 .
  • B The function is discontinuous at 𝑥 = 5 because l i m 𝑥 → 5 𝑓 ( 𝑥 ) ≠ 𝑓 ( 5 ) .
  • C The function is discontinuous at 𝑥 = 5 because 𝑓 ( 5 ) is undefined.
  • D The function is discontinuous at 𝑥 = 5 because l i m 𝑥 → 5 𝑓 ( 𝑥 ) does not exist.

Q8:

Find the value of 𝑎 that makes 𝑓 continuous at 𝑥 = 3 , given that

Q9:

Setting 𝑓 ( 𝑎 ) = 5 4 and 𝑓 ( 𝑥 ) = 𝑥 − 𝑎 𝑥 − 𝑎 6 6 3 3 when 𝑥 ≠ 𝑎 makes 𝑓 continuous at 𝑥 = 𝑎 . Determine 𝑎 .

  • A2
  • B 1 3
  • C 1 2
  • D3

Q10:

Consider the function 𝑓 ( 𝑥 ) = 1 − 𝑥 w h e n 𝑥 < 0 , 0 w h e n 𝑥 = 0 , 1 + 2 𝑥 w h e n 𝑥 > 0 .

What is 𝑓 ( 0 ) ?

What is l i m → 𝑓 ( 𝑥 ) ?

What is l i m → 𝑓 ( 𝑥 ) ?

What type of discontinuity does the function 𝑓 have at 𝑥 = 0 ?

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

Q11:

Consider the function

What is 𝑓 ( 0 ) ?

What is l i m 𝑥 → 0 − 𝑓 ( 𝑥 ) ?

  • AThe limit does not exist.
  • B − ∞
  • C + ∞
  • D0
  • E1

What is l i m 𝑥 → 0 + 𝑓 ( 𝑥 ) ?

  • A1
  • B0
  • C + ∞
  • DThe limit does not exist.
  • E − ∞

What type of discontinuity does the function 𝑓 have at 𝑥 = 0 ?

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

Q12:

Determine the value of 𝑎 that makes 𝑓 continuous at 𝑥 = 0 , given

  • A − 1 0
  • B 5 3
  • C 3 5 3
  • D − 6 5 3
  • E − 5 3

Q13:

Discuss the continuity of the function 𝑓 at 𝑥 = 0 , given

  • AThe function is discontinuous at 𝑥 = 0 because l i m 𝑥 → 0 𝑓 ( 𝑥 ) does not exist.
  • BThe function is discontinuous at 𝑥 = 0 because 𝑓 ( 0 ) is undefined.
  • CThe function is continuous at 𝑥 = 0 .
  • DThe function is discontinuous at 𝑥 = 0 because l i m 𝑥 → 0 𝑓 ( 𝑥 ) ≠ 𝑓 ( 0 ) .

Q14:

The function is continuous at 𝑥 = 0 . Determine the possible values of 𝑎 .

  • A √ 3 , − √ 3
  • B √ 3
  • C 2 , − 2
  • D 3 , − 3

Q15:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥 = 𝜋 4 , given

  • A − 1 6
  • B 1 2
  • C2
  • D 1 6

Q16:

Find the value of 𝑘 that makes the function 𝑓 continuous at 𝑥 = 𝜋 4 , given that

  • A − 6 5
  • B − 2
  • C − 6
  • D − 2 5

Q17:

Discuss the continuity of the function 𝑓 at 𝑥 = 𝜋 2 , given

  • AThe function is discontinuous at 𝑥 = 𝜋 2 because l i m 𝑥 → 𝜋 2 𝑓 ( 𝑥 ) does not exist.
  • BThe function is discontinuous at 𝑥 = 𝜋 2 because 𝑓 𝜋 2 is undefined.
  • CThe function is discontinuous at 𝑥 = 𝜋 2 because l i m 𝑥 → 𝜋 2 𝑓 ( 𝑥 ) ≠ 𝑓 𝜋 2 .
  • DThe function is continuous at 𝑥 = 𝜋 2 .
  • EThe function is discontinuous on ℝ .

Q18:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥 = 0 , given that

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

Q19:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥 = 0 , given that

  • A2
  • B 1 2
  • C 8 7
  • D 8 4 9
  • E 2 4 9

Q20:

Discuss the continuity of the function 𝑓 at 𝑥 = 𝜋 2 , given

  • AThe function is discontinuous at 𝑥 = 𝜋 2 because l i m 𝑥 → 𝜋 2 𝑓 ( 𝑥 ) does not exist.
  • BThe function is discontinuous at 𝑥 = 𝜋 2 because 𝑓 𝜋 2 is undefined.
  • CThe function is discontinuous at 𝑥 = 𝜋 2 because l i m 𝑥 → 𝜋 2 𝑓 ( 𝑥 ) ≠ 𝑓 𝜋 2 .
  • DThe function is continuous at 𝑥 = 𝜋 2 .

Q21:

Let Find all values of that make continuous at .

  • A
  • B
  • C
  • D

Q22:

Let Find all values of 𝑘 that make 𝑓 continuous at 𝑥 = 0 .

  • A 𝑘 ∈ ℝ
  • B0
  • C 𝑘 ∈ ℝ +
  • D 𝑘 ∈ ℝ − { 0 }

Q23:

Suppose What can be said of the continuity of 𝑓 at 𝑥 = 0 ?

  • A The function is discontinuous at 𝑥 = 0 because l i m 𝑥 → 0 𝑓 ( 𝑥 ) does not exist.
  • B The function is continuous on ℝ .
  • C The function is discontinuous at 𝑥 = 0 because 𝑓 ( 0 ) is undefined.
  • D The function is continuous at 𝑥 = 0 .
  • E The function is discontinuous at 𝑥 = 0 because 𝑓 ( 0 ) ≠ 𝑓 ( 𝑥 ) l i m 𝑥 → 0 .

Q24:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥 = 4 , given

  • A4
  • B − 1 4
  • C − 4
  • D 1 4
  • E − 1

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