5. A function is said to be continuous on
the interval when the function is
continuous at a number in that
interval. If c is a number in the
interval(a, b) and f is a function with
a domain containing the interval(a,b),
then f is said to be continuous at x = c
if all the following conditions are
satisfied:
12. Solution:
i. f (3)=
𝒙𝟐−𝟗
𝒙−𝟑
ii. lim
𝑥→3
𝒙𝟐−𝟗
𝒙−𝟑
iii. lim
𝑥→3
f(x) = f(x)
The three function are not satisfied.
The function is discontinuous at x = 3
13. Solution:
i. f (3)= −𝟐𝒙 + 𝟒
ii. lim
𝑥→3+
− 𝟐𝒙 + 𝟒
lim
𝑥→3−
(𝒙 − 𝟏)
iii. lim
𝑥→3
f(x) ≠ f(x)
The second condition is not satisfied at x = 3.
The function is discontinuous at x = 3
14. Solution:
i. f (-1)= 3
ii. lim
𝑥→−1+
+ (𝑥 + 1)
lim
𝑥→−1−
− (𝑥 + 1)
iii. lim
𝑥→−1
f(x) ≠ f(-1)
The function is discontinuous at x = -1
15. Solution:
i. f (2)= 4𝑥 − 2
ii. lim
𝑥→2+
(−2𝑥 + 5)
lim
𝑥→2−
(4𝑥 − 2)
iii. lim
𝑥→2+
f(x) ≠ lim
𝑥→2
f(x)
The function is discontinuous at x = 2
23. A removable discontinuity
if the limit of f(x) as x
approaches a exists, and not
equal to f(a).
lim
𝑥→𝑎
𝑓 𝑥 ≠ 𝑓(𝑥)
24. A jump discontinuity if the
limit of f(x) as x approaches to
a from the right is not equal to
the limit of f(x) as x
approaches to a from the left.
lim
𝑥→𝑎−
𝑓 𝑥 ≠ lim
𝑥→𝑎+
𝑓 𝑥
27. i. 𝑓 𝑎 exists.
The function is defined at a
The graph of the function contains 𝑎, 𝑓 𝑎
Examples of
functions not
continuous at
some x = a
28. ii. lim
𝑥→𝑎
𝑓 𝑥 exists
lim
𝑥→𝑎−
𝑓 𝑥 = lim
𝑥→𝑎+
𝑓 𝑥
Examples of
functions not
continuous at some
x = a
29. iii. lim
𝑥→𝑎
𝑓 𝑥 = 𝑓 𝑎
Example of functions which are not continuous at
some point x = a
30. A function which is not continuous at x = a is
discontinuous at that point.
31.
32. Graphically, a function is continuous in an
interval when its graph has no “breaks” or
“jumps”.
A function is continuous when one can trace its
graph without lifting the pencil from the paper.
38. If a function has a removable discontinuity at a
point a, that discontinuity can be removed by
redefining the function to fit continuity, in
particular, by making lim
𝑥→𝑎
𝑓 𝑥 = 𝑓 𝑎
41. If the discontinuities cannot be removed, the
discontinuity is called essential discontinuity.
i. f(2) is undefined. The
function is discontinuous
at 2
ii. lim
𝑥→2
𝑓 𝑥 exists
iii. lim
𝑥→2
𝑓 𝑥 ≠ 𝑓 2
42. i. 𝑓 2 =?, f(2) does not exist
f is discontinuous at x = 2
ii. lim
𝑥→2−
𝑓 𝑥 = lim
𝑥→𝑎+
𝑓 𝑥 = 4
iii. lim
𝑥→2
𝑓 𝑥 ≠ 𝑓 2