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AP Calculus AB: 2.1.8 Continuity and Discontinuity

Mathematics13 CardsCreated 3 months ago

A function is continuous at a point if it’s defined there, its limit exists, and the function’s value equals that limit. Discontinuities occur when one or more of these conditions fail, leading to jump, point (removable), or infinite discontinuities.

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Continuity and Discontinuity

  • 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.

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Key Terms

Term
Definition

Continuity and Discontinuity

  • A function is continuous at a point if it has no breaks or holes at that location.

  • Three conditions must be met for a functi...

note

  • Some functions behave exactly how you expect them to. Others jump around, have points in odd places, and generally behave strangely. If the...

note 2

  • There are two ways a function can be discontinuous.

  • The first way is called a jump discontinuity, or a break. Jump discontin...

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.

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

Suppose f(x)=x^2−1/x−1.

Which conditions of continuity are not met by f (x) at x = 1?

f(c) must be defined.

lim x→cf(x) must exist.

lim x→cf(x) = f(c).

Conditions 1 and 3.

Suppose f(x)=x^2−1/x−1.

Is f(x) continuous at x=1?

No, f (x) is not continuous at x = 1.

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AP Calculus AB: 2.1.8 Continuity and Discontinuity
TermDefinition

Continuity and Discontinuity

  • 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.

note

  • 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.

note 2

  • 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.

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.

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

Suppose f(x)=x^2−1/x−1.

Which conditions of continuity are not met by f (x) at x = 1?

f(c) must be defined.

lim x→cf(x) must exist.

lim x→cf(x) = f(c).

Conditions 1 and 3.

Suppose f(x)=x^2−1/x−1.

Is f(x) continuous at x=1?

No, f (x) is not continuous at x = 1.

Suppose g(x)=x^2−4/x+1.

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

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

Suppose f(x)={x + 1, x ≤ 0

−x + 1, x > 0

Is f(x) continuous at x=0?

Yes, f (x) is continuous at x = 0.

Classify all of the discontinuities of the function.

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

x = −2; infinite discontinuity

x = −1; infinite discontinuity

x = 1; removable (point) discontinuity

x = 4; removable (point) discontinuity

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

x+7,x>0.

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

Suppose f(x)={x^2−2, x≠2

0, x=2.

Condition three:

lim x→c f(x) = f(c).

Suppose f(x)={x+2, x < 3

x^2+1, x > 3.

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

Suppose f(x)=x^2−x−6/x+2.

Which statement describes the continuity of f at x = 3?

f is continuous at x = 3.