Worksheet: Continuity of Functions

In this worksheet, we will practice checking the continuity of a function over its domain and determining the interval on which it is continuous.

Q1:

Discuss the continuity of the function 𝑓 given 𝑓(𝑥)=𝑥4𝑥+2.

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

Q2:

Discuss the continuity of the function 𝑓(𝑥)=4.

  • A 𝑓 ( 𝑥 ) is continuous on {4}.
  • B 𝑓 ( 𝑥 ) is continuous on because it is a constant function.

Q3:

What can be said about the continuity of the function 𝑓(𝑥)=𝑥+5𝑥2𝑥+2?

  • A 𝑓 ( 𝑥 ) is discontinuous on because it is a polynomial.
  • B 𝑓 ( 𝑥 ) is continuous on because it is a polynomial.

Q4:

What can be said about the continuity of the function 𝑓(𝑥)=9𝑥+4𝑥cos?

  • A 𝑓 ( 𝑥 ) is continuous on because 𝑥𝑥cos is continuous on .
  • B 𝑓 ( 𝑥 ) is continuous on because 𝑥9𝑥 is a polynomial, and 𝑥𝑥cos is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on because 𝑥9𝑥 is a polynomial.

Q5:

Find the set on which 𝑓(𝑥)=(5𝑥5)5𝑥cos is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on because 𝑥(5𝑥5) is a polynomial.
  • B 𝑓 ( 𝑥 ) is continuous on because 𝑥(5𝑥)cos is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on because 𝑥(5𝑥5) is a polynomial, and 𝑥(5𝑥)cos is continuous on .
  • D 𝑓 ( 𝑥 ) is continuous on {5}.

Q6:

Find the set on which 𝑓(𝑥)=3(𝑥)+8tan is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on 𝑥𝑥=𝜋2+𝑛𝜋,𝑛.
  • B 𝑓 ( 𝑥 ) is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on {𝑥𝑥=𝑛𝜋,𝑛}.
  • D 𝑓 ( 𝑥 ) is continuous on 𝑥𝑥=𝑛𝜋2,𝑛.

Q7:

Find the set on which 𝑓(𝑥)=4𝑥+10𝑥+9 is continuous.

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

Q8:

Find the set on which 𝑓(𝑥)=39𝑥+𝑥𝑥1sincoscos is continuous.

  • A { 2 𝜋 𝑛 , 𝑛 }
  • B
  • C 𝜋 2 + 𝑛 𝜋 , 𝑛
  • D { 1 }

Q9:

Discuss the continuity of the function 𝑓(𝑥)=2𝑥|𝑥9|+5 on .

  • A 𝑓 ( 𝑥 ) is continuous on .
  • B 𝑓 ( 𝑥 ) is discontinuous on .
  • C 𝑓 ( 𝑥 ) is continuous on {9}.
  • D 𝑓 ( 𝑥 ) is continuous on {5}.

Q10:

Find the set on which 𝑓(𝑥)=4|𝑥| is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on {4}.
  • B 𝑓 ( 𝑥 ) is continuous on [4,4].
  • C 𝑓 ( 𝑥 ) is continuous on (4,4).
  • D 𝑓 ( 𝑥 ) is continuous on .
  • E 𝑓 ( 𝑥 ) is continuous on (4,4).

Q11:

Find the set on which 𝑓(𝑥)=7𝑥|𝑥|8 is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on {8,8}.
  • B 𝑓 ( 𝑥 ) is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on {8}.
  • D 𝑓 ( 𝑥 ) is continuous on {8}.

Q12:

Find the set on which 𝑓(𝑥)=|𝑥14|(𝑥4) is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on {4}.
  • B 𝑓 ( 𝑥 ) is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on {14}.
  • D 𝑓 ( 𝑥 ) is continuous on {4}.

Q13:

Discuss the continuity of the function 𝑓, given 𝑓(𝑥)=3+𝑥0𝑥<𝜋2,4+𝑥𝜋2𝑥𝜋2.sinifif

  • A The function 𝑓 is continuous on 0,𝜋2.
  • B The function 𝑓 is continuous on 0,𝜋2.
  • C The function 𝑓 is continuous on 0,𝜋2.
  • D The function 𝑓 is continuous on (0,).
  • E The function 𝑓 is continuous on [0,).

Q14:

Discuss the continuity of the function 𝑓, given 𝑓(𝑥)=(8𝑥+40)𝑥5𝑥<5,8𝑥25𝑥5.sinifif

  • A The function 𝑓 is continuous on (,5].
  • B The function 𝑓 is continuous on {5,5}.
  • C The function 𝑓 is continuous on {5}.
  • D The function 𝑓 is continuous on [5,).
  • E The function 𝑓 is continuous on .

Q15:

Discuss the continuity if the function 𝑓 given 𝑓(𝑥)=3𝑥+258𝑥7,3𝑥167<𝑥<3,9𝑥243𝑥<9.ififif

  • A 𝑓 ( 𝑥 ) is continuous on .
  • B 𝑓 ( 𝑥 ) is continuous on [8,9).
  • C 𝑓 ( 𝑥 ) is continuous on (8,9).
  • D 𝑓 ( 𝑥 ) is continuous on [8,9){7,3}.

Q16:

If 𝑓(𝑥)=𝑥𝑥+𝑥𝑥𝜋4<𝑥<0,24𝑥0𝑥<𝜋4,tansinifcosif what can be said of the continuity of 𝑓 on the interval 𝜋4,𝜋4?

  • A 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4{0}.
  • B 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4{1}.
  • C 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4.
  • D 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4{0,1}.

Q17:

If 𝑓(𝑥)=𝑥𝑥+𝑥𝑥𝜋4<𝑥<0,64𝑥0𝑥<𝜋4,tansinifcosif what can be said of the continuity of 𝑓 on the interval 𝜋4,𝜋4?

  • A 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4𝜋8,𝜋8.
  • B 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4.
  • C 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋4{0}.
  • D 𝑓 ( 𝑥 ) is continuous on 𝜋4,𝜋40,𝜋8.

Q18:

Which of the following holds for 𝑓(𝑥)=|𝑥+14|+|8+𝑥|?

  • A 𝑓 ( 𝑥 ) is continuous on (8,).
  • B 𝑓 ( 𝑥 ) is continuous on {14,8}.
  • C 𝑓 ( 𝑥 ) is continuous on (,14).
  • D 𝑓 ( 𝑥 ) is continuous on because it is a polynomial of first degree.
  • E 𝑓 ( 𝑥 ) is continuous on {14}.

Q19:

Suppose 𝑓(𝑥)=5𝑥|𝑥4|4𝑥𝑥<4,|7𝑥9|𝑥4.ifif Find the set on which 𝑓 is continuous.

  • A The function 𝑓(𝑥) is continuous on .
  • B The function 𝑓(𝑥) is continuous on (4,).
  • C The function 𝑓(𝑥) is continuous on {4}.
  • D The function 𝑓(𝑥) is continuous on (,4).

Q20:

Find the set on which 𝑓(𝑥)=9𝑥|𝑥|22𝑥0,472|𝑥|𝑥𝑥>0ifif is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on (0,).
  • B 𝑓 ( 𝑥 ) is continuous on (,0).
  • C 𝑓 ( 𝑥 ) is continuous on {0}.
  • D 𝑓 ( 𝑥 ) is continuous on .

Q21:

Determine where 𝑓 is continuous if 𝑓(𝑥)=2𝑥𝜋4𝑥<3𝜋4,3𝑥3𝜋4𝑥2𝜋.sinifcosif

  • A 𝑓 ( 𝑥 ) is continuous on 𝜋4,2𝜋.
  • B 𝑓 ( 𝑥 ) is continuous on .
  • C 𝑓 ( 𝑥 ) is continuous on 3𝜋4.
  • D 𝑓 ( 𝑥 ) is continuous on 𝜋4,2𝜋.
  • E 𝑓 ( 𝑥 ) is continuous on 𝜋4,2𝜋3𝜋4.

Q22:

Discuss the continuity of the function 𝑓 on , given 𝑓(𝑥)=75𝑥𝜋7𝑥<𝜋4,53𝑥𝜋4𝑥2𝜋.sinifcosif

  • A 𝑓 ( 𝑥 ) is continuous on 𝜋7,𝜋4.
  • B 𝑓 ( 𝑥 ) is continuous on 𝜋7,2𝜋𝜋4.
  • C 𝑓 ( 𝑥 ) is continuous on (,2𝜋].
  • D 𝑓 ( 𝑥 ) is continuous on 𝜋4.
  • E 𝑓 ( 𝑥 ) is continuous on 𝜋7,.

Q23:

Discuss the continuity of the function 𝑓, given 𝑓(𝑥)=7𝑥𝜋6<𝑥0,7𝑥𝑥0<𝑥<𝜋6.cosiftanif

  • A The function 𝑓 is continuous on 𝜋6,𝜋6.
  • B The function 𝑓 is continuous on 0,𝜋6.
  • C The function 𝑓 is continuous on 𝜋6,0.
  • D The function 𝑓 is continuous on 𝜋6,𝜋6.
  • E The function 𝑓 is continuous on .

Q24:

Find the value of 𝑎 that makes the function 𝑓 continuous given 𝑓(𝑥)=8𝑥+56𝑥2𝜋𝑥<0,𝑎+4𝑥0𝑥𝜋.sinsinifif

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

Q25:

Discuss the continuity of the function 𝑓, given 𝑓(𝑥)=2𝑥+2𝑥𝜋6<𝑥<0,57+8𝑥0𝑥<𝜋6.sinifcosif

  • A The function 𝑓 is continuous on {0}.
  • B The function 𝑓 is continuous on 𝜋6,𝜋6.
  • C The function 𝑓 is continuous on 𝜋6,𝜋6{0}.
  • D The function 𝑓 is continuous on 0,𝜋6.
  • E The function 𝑓 is continuous on 𝜋6,𝜋6.

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