AP Calculus BC Flashcards: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
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State the theorem connecting differentiability to continuity.
The theorem states that if a function is differentiable at a point, then it is also continuous at that point.
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State the theorem connecting differentiability to continuity.
The theorem states that if a function is differentiable at a point, then it is also continuous at that point.
True or False: A function can be differentiable at x=a but not continuous at x=a.
False. If a function is differentiable at a point, it must be continuous there.
What is the fundamental relationship between differentiability and continuity?
If a function is differentiable at a point, then it must also be continuous at that point. However, a continuous function is not necessarily differentiable.
If a function is continuous at a point, is it guaranteed to be differentiable at that point?
No, a continuous function may fail to be differentiable at a point in its domain.
Differentiability implies...?
Differentiability at a point implies continuity at that point.
A function g(x) is continuous everywhere on its domain. Can you conclude that g'(x) exists everywhere on its domain?
No, you cannot. A continuous function may have points where it is not differentiable.
Describe the 'one-way' nature of the relationship between differentiability and continuity.
The relationship is 'one-way' because differentiability implies continuity, but the converse (continuity implying differentiability) is not true.
True or False: A function can be continuous at x=a but not differentiable at x=a.
True. A continuous function can fail to be differentiable at a point within its domain.
Which is a stronger condition for a function at a point: differentiability or continuity?
Differentiability is a stronger condition because it guarantees continuity, while continuity does not guarantee differentiability.
Given that f'(c) exists, what can you definitively conclude about the function f(x) at x=c?
You can definitively conclude that the function f(x) is continuous at x=c.