2.1.8 Continuity and Discontinuity

Question 1

Continuity and Discontinuity

Answer 1

• A function is continuous at a point if it has no breaks or holes at that location.
• Three conditions must be met for a function to be continuous at a point.

Question 2

note

Answer 2

• Some functions behave exactly how you expect them to. Others jump around, have points in odd places, and generally behave strangely. If the curve of a function is well behaved at a given point, then the function is said to be continuous at that point. Otherwise the function is discontinous at that point.
• Three conditions must be met for a function to be continuous at a point.
  1. The function must be defined at that point.
  2. The limit of the function at that point must exist.
  3. The function and the limit must be equal.
• Although continuity is defined point by point, if a curve is continuous for all values then it is okay to say that the function itself is continuous.

Question 3

note 2

Answer 3

• There are two ways a function can be discontinuous.
• The first way is called a jump discontinuity, or a break. Jump discontinuities occur when the left-handed and right-handed limits do not agree with each other.
• The greatest integer function is an example of a function with jump discontinuities. Look for jump discontinuities any time you work with piecewise-defined functions.
• The second type of discontinuity is a point discontinuity, or a hole. Point discontinuities occur when the limit exists but disagrees with the function.
• Point discontinuities are often seen when dealing with rational functions. Look for point discontinuities when dealing with piecewise-defined functions as well.

Question 4

Classify all of the discontinuities of the function.
f(x)=[x]+x,
−1 ≤ x ≤ 2, x≠3/2

Hint: “[x]” denotes the greatest integer function.

Answer 4

x=0; jump discontinuity
x=1; jump discontinuity
x=3/2; removable (point) discontinuity
x=2; jump discontinuity

Question 5

Suppose f(x)=x^2−1/x−1.
Which conditions of continuity are not met by f (x) at x = 1?

1. f(c) must be defined.
2. lim x→cf(x) must exist.
3. lim x→cf(x) = f(c).

Answer 5

Conditions 1 and 3.

Question 6

Suppose f(x)=x^2−1/x−1.
Is f(x) continuous at x=1?

Answer 6

No, f (x) is not continuous at x = 1.

Question 7

Suppose g(x)=x^2−4/x+1.

Is the function g continuous on the interval [−2, 2]?

Answer 7

g has a non-removable discontinuity on the given interval.

Question 8

Suppose f(x)={x + 1, x ≤ 0
                      −x + 1, x > 0
Is f(x) continuous at x=0?

Answer 8

Yes, f (x) is continuous at x = 0.

Question 9

Classify all of the discontinuities of the function.

f(x)=(x−1)(x+3)(x)/(x−1)(x+1)(x+2), x≠4

Answer 9

x = −2; infinite discontinuity
x = −1; infinite discontinuity
x = 1; removable (point) discontinuity
x = 4; removable (point) discontinuity

Question 10

Suppose f(x)={x^2+7,x<0
                       x+7,x>0.

Which statement describes the continuity of f at x = 0?

Answer 10

f has a point discontinuity (removable discontinuty) at x = 0.

Question 11

Suppose f(x)={x^2−2, x≠2
                       0, x=2.

Which condition of continuity is not met by f (x) at x = 2?

Answer 11

Condition three:

lim x→c f(x) = f(c).

Question 12

Suppose f(x)={x+2, x < 3
                       x^2+1, x > 3.

Is f continuous on the interval [−2, 2]?

Answer 12

Yes, f is continuous on the interval [−2, 2].

Question 13

Suppose f(x)=x^2−x−6/x+2.
Which statement describes the continuity of  f  at x = 3?

Answer 13

f is continuous at x = 3.