Differential Calculus 201-NYA-05
Vincent Carrier
Exercise Sheet 7
2.5 Continuity
y
b 4
6
3
r
r
2
r
r 1
0 1
−4 −3 −2 −1
b
−1
b
b
2
3
4 x
−2
−3
−4
1. Find the points where the above function is not continuous and identify the type of
discontinuity for each.
For numbers 2 to 13, find the point(s) at which the function is not continuous (if there
are any) and identify for each the type of discontinuity.
2. f (x) =
x
2
x −9
3. f (x) =
x2 − 3x − 4
x−4
4. f (x) =
6 − 2x
|3 − x|
5. f (x) =
x2 + 3x − 4
x2 − 1
6. f (x) =
x3 − 3x + 2
x2 + x + 2
7. f (x) =
x2 − 2x − 3
x2 − x − 2
 2

 x +x−6
x−2
8. f (x) =

 x2 + 4x − 9


x2 − 5x − 9




2
10. f (x) =



x2 − 9x + 14


7−x
if x < 2
if x ≥ 2
 2
−1

 (x + 1) − 1/2
x+1
9. f (x) =


1
if x < 7
if x = 7
if x > 7
11. f (x) =
 2
x + 5x + 4


√


 3 − 1 − 2x





if x 6= −1
if x = −1
if x < −4
−9
if x = −4
7 − x2
if x > −4
12. f (x) =
 2
x + 2x − 3




 (9 − x2 )8







x2 − 4x − 2




4
13. f (x) =

 x2 − 7x + 10



|5 − x|
if x < −3
4
if x = −3
x3 + 2x2 + 1
if x > −3
if x < 5
if x = 5
if x > 5
Answers:
1. removable: x = −2
2.
x = −3, 3
3.
x=4
4.
x=3
5.
x=1
x = −1
6.
Continuous on R
7.
x = −1
x=2
8.
x=2
9.
x = −1
10.
x=7
jump: x = −3, −1, 1
lim
x→−3−
x = −3
13.
x=5
= −∞, lim− = −∞
infinite
x→3
lim f (x) = 5
removable
x→4
lim f (x) = 2, lim+ f (x) = −2
jump
lim f (x) = 5/2
removable
x→3−
x→3
x→1
lim f (x) = −∞
infinite
lim f (x) = 4/3
removable
lim f (x) = ∞
infinite
x→−1−
x→−1
x→2−
lim f (x) = 5, lim f (x) = 3, f (2) = 3
jump
lim f (x) = 1/2, f (−1) = 1
removable
lim f (x) = 5, lim f (x) = −5, f (7) = 2
jump
x→2−
x→2+
x→−1
x→7−
x→7+
lim f (x) = −9,
11. Continuous on R
12.
infinite: x = 2
x→−4−
lim f (x) = ∞,
x→−3−
lim f (x) = −9, f (−4) = −9
x→−4+
lim f (x) = −8, f (−3) = 4
x→−3+
lim f (x) = 3, lim f (x) = 3, f (5) = 4
x→5−
x→5+
infinite
removable