# AP Calculus AB : Relationship between differentiability and continuity

## Example Questions

### Example Question #1 : Relationship Between Differentiability And Continuity

The function  is differentiable at the point . List which of the following statements must be true about :

1)   The limit     exists.

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2)

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3)

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4)

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5)

1,3,4,and 5

1, 2, and 4

1, and 5

All must be true.

1, 3, and 5

1, 2, and 4

Explanation:

1) If a function is differentiable, then by definition of differentiability the limit defined by,

exists. Therefore (1) is required by definition of differentiability. _______________________________________________________________

2) If a function is differentiable at a point then it must also be continuous at that point. (This is not conversely true).

For a function to be continuous at a point  we must have:

Therefore (2) and (4) are required.

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3)

This is not required, the left side of the equation is the definition of a derivative at a point  for a function . The derivative at a point does not have to equal to the function value  at that point, it is equal to the slope  at that point. Therefore 3 does not have to be true.

However, we can note that it is possible for a function and its' derivative to be equal for a given point. Sine and cosine, for instance will intersect periodically. Another example would be the exponential function  which has itself as its' derivative

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4) See 2

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5)

Again, the function does not have to approach the same limit as its' derivative. It is possible for a function to behave in this manner, such as in the case of sine and its' derivative cosine, which will both have the same limit at points where they intersect.

### Example Question #1 : Concept Of The Derivative

When the limit  fails to exist,

The function  is not continuous at .

The function  is not defined at .

None of the above necessarily

The function  is not differentiable at .

The function  is not differentiable at .

Explanation:

By definition of differentiability,  when the limit exists. When  exists, we say the function is 'differentiable at '.

### Example Question #3 : Concept Of The Derivative

Which of the following functions is differentiable at ,  but not continuous there?

They are all differentiable and continuous at

They are all differentiable and continuous at

Explanation:

All of the functions are differentiable at . If you examine the graph of each of the functions, they are all defined at , and do not have a corner, cusp, or a jump there; they are all smooth and connected (Not necessarily everywhere, just at ). Additionally it is not possible to have a function that is differentiable at a point, but not continuous at that same point; differentiablity implies continuity.

### Example Question #4 : Concept Of The Derivative

For which of the following functions does a limit exist at , but not a y-value?

Explanation:

To answer the question, we must find an equation which satisfies two criteria:

(1) it must have limits on either side of  that approach the same value and (2) it must have a hole at .

Each of the possible answers provide situations which demonstrate each combination of (1) and (2). That is to say, some of the equations include both a limit and a y-value at neither, or,in the case of the piecewise function, a y-value and a limit that does not exist.

In the function, , the numerator factors to

while the denominator factors to . As a result, the graph of this

function resembles that for , but with a hole at . Therefore, the limit

at  exists, even though the y-value is undefined at .