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Worksheet: Continuity over an Interval

Q1:

Discuss the continuity of the function 𝑓 ( π‘₯ ) = 4 .

  • A 𝑓 ( π‘₯ ) is continuous on ℝ because it is a constant function.
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 4 } .

Q2:

What can be said about the continuity of the function 𝑓 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 2 π‘₯ + 2 3 2 ?

  • A 𝑓 ( π‘₯ ) is continuous on ℝ because it is a polynomial.
  • B 𝑓 ( π‘₯ ) is discontinuous on ℝ because it is a polynomial.

Q3:

Discuss the continuity of the function 𝑓 given 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 2 2 .

  • A The function is continuous on ℝ βˆ’ { βˆ’ 2 } .
  • B The function is continuous on ℝ .

Q4:

What can be said about the continuity of the function 𝑓 ( π‘₯ ) = 9 π‘₯ + 4 π‘₯ 2 2 c o s ?

  • A 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ ↦ π‘₯ c o s 2 is continuous on ℝ .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ ↦ 9 π‘₯ 2 is a polynomial.
  • C 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ ↦ 9 π‘₯ 2 is a polynomial, and π‘₯ ↦ π‘₯ c o s 2 is continuous on ℝ .

Q5:

Find the set on which 𝑓 ( π‘₯ ) = ( 5 π‘₯ βˆ’ 5 ) 5 π‘₯ c o s is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ β†’ ( 5 π‘₯ βˆ’ 5 ) is a polynomial.
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 5 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ β†’ ( 5 π‘₯ ) c o s is continuous on ℝ .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ because π‘₯ β†’ ( 5 π‘₯ βˆ’ 5 ) is a polynomial, and π‘₯ β†’ ( 5 π‘₯ ) c o s is continuous on ℝ .

Q6:

Find the set on which 𝑓 ( π‘₯ ) = βˆ’ 3 ( π‘₯ ) + 8 t a n 2 is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = 𝑛 πœ‹ , 𝑛 ∈ β„€ } .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = 𝑛 πœ‹ 2 , 𝑛 ∈ β„€  .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = πœ‹ 2 + 𝑛 πœ‹ , 𝑛 ∈ β„€  .

Q7:

Find the set on which 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + 1 0 π‘₯ + 9 βˆ’ 3 2 is continuous.

  • AThe function 𝑓 is continuous on ℝ βˆ’ { 0 , βˆ’ 3 } .
  • BThe function 𝑓 is continuous on ℝ .
  • CThe function 𝑓 is continuous on ℝ βˆ’ { βˆ’ 3 } .
  • DThe function 𝑓 is continuous on ℝ βˆ’ { 0 } .
  • EThe function 𝑓 is continuous on ℝ βˆ’ { 0 , 3 } .

Q8:

Find the set on which 𝑓 ( π‘₯ ) = βˆ’ 3 9 π‘₯ + π‘₯ π‘₯ βˆ’ 1 s i n c o s c o s is continuous.

  • A ℝ βˆ’ { 1 }
  • B ℝ
  • C ℝ βˆ’  πœ‹ 2 + 𝑛 πœ‹ , 𝑛 ∈ β„€ 
  • D ℝ βˆ’ { 2 πœ‹ 𝑛 , 𝑛 ∈ β„€ }

Q9:

Discuss the continuity of the function 𝑓 ( π‘₯ ) = 2 π‘₯ | π‘₯ βˆ’ 9 | + 5 on ℝ .

  • A 𝑓 ( π‘₯ ) is discontinuous on ℝ .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 9 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 5 } .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ .

Q10:

Find the set on which is continuous.

  • A is continuous on
  • B is continuous on
  • C is continuous on
  • D is continuous on
  • E is continuous on

Q11:

Find the set on which 𝑓 ( π‘₯ ) = 7 π‘₯ | π‘₯ | βˆ’ 8 is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { βˆ’ 8 } .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 8 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 8 , βˆ’ 8 } .

Q12:

Find the set on which 𝑓 ( π‘₯ ) = | π‘₯ βˆ’ 1 4 | ( π‘₯ βˆ’ 4 ) 2 is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 1 4 } .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { βˆ’ 4 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 4 } .

Q13:

Discuss the continuity of the function , given

  • A The function is continuous on .
  • B The function is continuous on .
  • C The function is continuous on .
  • D The function is continuous on .
  • E The function is continuous on .

Q14:

Discuss the continuity of the function , given

  • A The function is continuous on .
  • B The function is continuous on .
  • C The function is continuous on .
  • D The function is continuous on .
  • E The function is continuous on .

Q15:

Discuss the continuity if the function given

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .

Q16:

If what can be said of the continuity of on the interval ?

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .

Q17:

If what can be said of the continuity of on the interval ?

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .

Q18:

Which of the following holds for ?

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on because it is a polynomial of first degree.
  • E is continuous on .

Q19:

Suppose Find the set on which is continuous.

  • A The function is continuous on .
  • B The function is continuous on .
  • C The function is continuous on .
  • D The function is continuous on .

Q20:

Find the set on which is continuous.

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .

Q21:

Determine where is continuous if

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .
  • E is continuous on .

Q22:

Discuss the continuity of the function on , given

  • A is continuous on .
  • B is continuous on .
  • C is continuous on .
  • D is continuous on .
  • E is continuous on .

Q23:

Discuss the continuity of the function , given

  • A The function is continuous on .
  • B The function is continuous on .
  • C The function is continuous on .
  • D The function is continuous on .
  • E The function is continuous on .

Q24:

Find the value of π‘Ž that makes the function 𝑓 continuous given

  • A βˆ’ 1 3 2
  • B76
  • C 1 3 2
  • D βˆ’ 7 6
  • E βˆ’ 1 9

Q25:

Discuss the continuity of the function , given

  • A The function is continuous on .
  • B The function is continuous on .
  • C The function is continuous on .
  • D The function is continuous on .
  • E The function is continuous on .