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 𝑓(𝑥)=𝑥+8𝑥−4𝑥≠−2,−3𝑥=−2.ifif

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

Q2:

Given 𝑓(𝑥)=−7𝑥+8𝑥<−8,𝑥+2𝑥+4𝑥>−8.ifif If possible or necessary, define 𝑓(−8) so that 𝑓 is continuous at 𝑥=−8.

  • AThe function is already continuous at 𝑥=−8.
  • B𝑓(−8)=0 would make 𝑓 continuous at 𝑥=−8.
  • C𝑓(−8)=6 would make 𝑓 continuous at 𝑥=−8.
  • DThe function cannot be made continuous at 𝑥=−8 because lim→𝑓(𝑥)≠lim→𝑓(𝑥).

Q3:

Suppose 𝑓(𝑥)=⎧⎨⎩𝑥+1𝑥−1𝑥<−1,62𝑥−10𝑥≥−1.ifif What can be said of the continuity of 𝑓 at 𝑥=−1?

  • AThe function is continuous on ℝ.
  • BThe function is discontinuous at 𝑥=−1 because 𝑓(−1) is undefined.
  • CThe function is continuous at 𝑥=−1.
  • DThe function is discontinuous at 𝑥=−1 because lim→𝑓(𝑥) does not exist.
  • EThe function is discontinuous at 𝑥=−1 because 𝑓(−1)≠𝑓(𝑥)lim→.

Q4:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥=−1 and 𝑥=−6, given that 𝑓(𝑥)=3𝑥+11𝑥≤−6,𝑎𝑥+𝑏−6<𝑥<−1,−5𝑥+10𝑥≥−1.ififif

  • A𝑎=125, 𝑏=375
  • B𝑎=375, 𝑏=125
  • C𝑎=−617, 𝑏=−127
  • D𝑎=−127, 𝑏=−617
  • E𝑎=125, 𝑏=−2575

Q5:

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

  • AThe function is already continuous at 𝑥=1.
  • BNo value of 𝑓(1) will make 𝑓 continuous because lim→𝑓(𝑥) does not exist.
  • C𝑓(1)=3 makes 𝑓 continuous at 𝑥=1.
  • DThe function cannot be made continuous at 𝑥=1 because 𝑓(1) is undefined.

Q6:

Discuss the continuity of the function 𝑓 at 𝑥=5 given 𝑓(𝑥)=⎧⎪⎨⎪⎩8𝑥+1𝑥≤5,𝑥−25𝑥−125𝑥>5.ifif

  • AThe function is discontinuous at 𝑥=5 because lim→𝑓(𝑥) does not exist.
  • BThe function is discontinuous at 𝑥=5 because 𝑓(5) is undefined.
  • CThe function is discontinuous at 𝑥=5 because lim→𝑓(𝑥)≠𝑓(5).
  • DThe function is continuous at 𝑥=5.

Q7:

Find the value of 𝑎 that makes 𝑓 continuous at 𝑥=3, given that 𝑓(𝑥)=𝑥+𝑥(𝑎−3)−3𝑎𝑥−3𝑥≠3,7𝑥+6𝑥=3.ifif

Q8:

Setting 𝑓(𝑎)=54 and 𝑓(𝑥)=𝑥−𝑎𝑥−𝑎 when 𝑥≠𝑎 makes 𝑓 continuous at 𝑥=𝑎. Determine 𝑎.

  • A12
  • B3
  • C13
  • D2

Q9:

Determine the value of 𝑎 that makes 𝑓 continuous at 𝑥=0, given 𝑓(𝑥)=−56𝑥−57𝑥3𝑥𝑥≠0,𝑎𝑥=0.sintanifif

  • A−653
  • B353
  • C−10
  • D−53
  • E53

Q10:

Discuss the continuity of the function 𝑓 at 𝑥=0, given 𝑓(𝑥)=𝑥5𝑥𝑥≠0,5𝑥=0.sinifif

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

Q11:

The function 𝑓(𝑥)=7|𝑥|𝑥+17𝑥<0,𝑎+9𝑥𝑥≥0ifcosif is continuous at 𝑥=0. Determine the possible values of 𝑎.

  • A3,−3
  • B√3,−√3
  • C√3
  • D2,−2

Q12:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥=𝜋4, given 𝑓(𝑥)=⎧⎨⎩2𝑥+9𝑥4+4𝑥𝑥≠𝜋4,3𝑎𝑥=𝜋4.sintansinifif

  • A12
  • B−16
  • C2
  • D16

Q13:

Find the value of 𝑘 that makes the function 𝑓 continuous at 𝑥=𝜋4, given that 𝑓(𝑥)=⎧⎨⎩−62𝑥+4𝑥2𝑥+2𝑥≠𝜋4,5𝑘𝑥=𝜋4.sintansinifif

  • A−2
  • B−25
  • C−65
  • D−6

Q14:

Discuss the continuity of the function 𝑓 at 𝑥=𝜋2, given 𝑓(𝑥)=−7𝑥+7𝑥𝑥≤𝜋2,62𝑥−1𝑥>𝜋2.sincosifcosif

  • AThe function is discontinuous at 𝑥=𝜋2 because 𝑓𝜋2 is undefined.
  • BThe function is continuous at 𝑥=𝜋2.
  • CThe function is discontinuous on ℝ.
  • DThe function is discontinuous at 𝑥=𝜋2 because lim→𝑓(𝑥) does not exist.
  • EThe function is discontinuous at 𝑥=𝜋2 because lim→𝑓(𝑥)≠𝑓𝜋2.

Q15:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥=0, given that 𝑓(𝑥)=68𝑥8𝑥2𝑥𝑥≠0,𝑎−6𝑥=0.sintanifif

  • A30
  • B198
  • C24
  • D9

Q16:

Discuss the continuity of the function 𝑓 at 𝑥=𝜋2, given 𝑓(𝑥)=8+7𝑥𝑥<𝜋2,7+5𝑥𝑥≥𝜋2.cosifsinif

  • AThe function is discontinuous at 𝑥=𝜋2 because lim→𝑓(𝑥)≠𝑓𝜋2.
  • BThe function is discontinuous at 𝑥=𝜋2 because 𝑓𝜋2 is undefined.
  • CThe function is continuous at 𝑥=𝜋2.
  • DThe function is discontinuous at 𝑥=𝜋2 because lim→𝑓(𝑥) does not exist.

Q17:

Let 𝑓(𝑥)=⎧⎨⎩𝜋𝑥5𝑥𝑥<0,𝜋𝑎+6𝜋5𝑥𝑥≥0.sinifcosif Find all values of 𝑎 that make 𝑓 continuous at 𝑥=0.

  • A15
  • B2𝜋5−1
  • C65
  • D−45

Q18:

Let 𝑓(𝑥)=−7𝑥+78𝑘𝑥𝑥≠0,−24𝑥𝑥=0.cosifsinif Find all values of 𝑘 that make 𝑓 continuous at 𝑥=0.

  • A𝑘∈ℝ
  • B0
  • C𝑘∈ℝ
  • D𝑘∈ℝ−{0}

Q19:

Discuss the continuity of the function 𝑓 at 𝑥=−7, given that 𝑓(𝑥)=|𝑥+7|𝑥≤−2,−𝑥+3𝑥>−2.ifif

  • AThe function is discontinuous at 𝑥=−7 because 𝑓(−7)≠𝑓(𝑥)lim→.
  • BThe function is discontinuous at 𝑥=−7 because 𝑓(−7) is undefined.
  • CThe function is continuous at 𝑥=−7.
  • DThe function is continuous at all points in ℝ−{−7}.
  • EThe function is discontinuous at 𝑥=−7 because lim→𝑓(𝑥) does not exist.

Q20:

Discuss the continuity of the function 𝑓 at 𝑥=0, given that 𝑓(𝑥)=⎧⎪⎨⎪⎩6𝑥−𝑥𝑥4𝑥𝑥<0,𝑥+5𝑥+4𝑥≥0.sintanifif

  • AThe function is discontinuous on ℝ.
  • BThe function is discontinuous at 𝑥=0 because lim→𝑓(𝑥) does not exist.
  • CThe function is continuous at 𝑥=0.
  • DThe function is discontinuous at 𝑥=0 because 𝑓(0) is undefined.
  • EThe function is discontinuous at 𝑥=0 because lim→𝑓(𝑥)≠𝑓(0).

Q21:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=0, given 𝑓(𝑥)=2𝑥3𝑥𝑥≠0,𝑘𝑥=0.sincotifif

  • A6
  • B2
  • C32
  • D23

Q22:

Given 𝑓(𝑥)=−6𝑥𝑥<0,−6𝑥+1−𝑥𝑥𝑥>0.cosifcosif If possible or necessary, define 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • AThe function cannot be made continuous at 𝑥=0 because 𝑓(0) is undefined.
  • B𝑓(0)=−6 makes 𝑓 continuous at 𝑥=0.
  • CNo value of 𝑓(0) will make 𝑓 continuous because lim→𝑓(𝑥) does not exist.
  • DThe function is already continuous at 𝑥=0.

Q23:

Discuss the continuity of the function 𝑓 at 𝑥=0 given 𝑓(𝑥)=5𝑥+7𝑥2𝑥𝑥≠0,6𝑥=0.sinsinifif

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

Q24:

Suppose 𝑓(𝑥)=⎧⎨⎩−4𝑥+9𝑥≤3𝜋2,(4𝑥−6𝜋)+13𝑥>3𝜋2.sinifif What can be said of the continuity of 𝑓 at 𝑥=3𝜋2?

  • AThe function is continuous at 𝑥=3𝜋2.
  • BThe function is discontinuous at 𝑥=3𝜋2 because 𝑓3𝜋2 is undefined.
  • CThe function is discontinuous at 𝑥=3𝜋2 because lim→𝑓(𝑥) does not exist.
  • DThe function is discontinuous at 𝑥=3𝜋2 because 𝑓3𝜋2≠𝑓(𝑥)lim→.

Q25:

Given 𝑓(𝑥)=4𝑥𝑥<0,−8𝑥−7𝑥𝑥>0.tanifsincosif If possible or necessary, define 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • A𝑓(𝑥)=0 would make 𝑓 continuous at 𝑥=0.
  • B𝑓(𝑥)=−7 would make 𝑓 continuous at 𝑥=0.
  • CThe function can not be made continuous at 𝑥=0 because lim→𝑓(𝑥) does not exist.
  • DThe function is already continuous at 𝑥=0.

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