2.1.8 Continuity and Discontinuity Flashcards Preview

AP Calculus AB > 2.1.8 Continuity and Discontinuity > Flashcards

Flashcards in 2.1.8 Continuity and Discontinuity Deck (13)
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1
Q

Continuity and Discontinuity

A
  • 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.
2
Q

note

A
  • 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.
3
Q

note 2

A
  • 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.
4
Q

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.

A

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

5
Q
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).
A

Conditions 1 and 3.

6
Q
Suppose f(x)=x^2−1/x−1.
Is f(x) continuous at x=1?
A

No, f (x) is not continuous at x = 1.

7
Q

Suppose g(x)=x^2−4/x+1.

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

A

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

8
Q
Suppose f(x)={x + 1, x ≤ 0
                      −x + 1, x > 0
Is f(x) continuous at x=0?
A

Yes, f (x) is continuous at x = 0.

9
Q

Classify all of the discontinuities of the function.

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

A
x = −2; infinite discontinuity
x = −1; infinite discontinuity
x = 1; removable (point) discontinuity
x = 4; removable (point) discontinuity
10
Q
Suppose f(x)={x^2+7,x<0
                       x+7,x>0.

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

A

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

11
Q
Suppose f(x)={x^2−2, x≠2
                       0, x=2.

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

A

Condition three:

lim x→c f(x) = f(c).

12
Q
Suppose f(x)={x+2, x < 3
                       x^2+1, x > 3.

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

A

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

13
Q
Suppose f(x)=x^2−x−6/x+2.
Which statement describes the continuity of  f  at x = 3?
A

f is continuous at x = 3.

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