Understanding Differentiability and Continuity

The video explores differentiability at a point, defining it as the existence of a derivative. It claims that if a function is differentiable at a point, it is also continuous there, but not vice versa. Examples of non-continuous functions, such as those with discontinuities, are discussed to show they are not differentiable. The video also covers removable discontinuities and continuous functions that are not differentiable, like the absolute value function, illustrating that continuity does not guarantee differentiability.

What is the primary focus of differentiability at a point?

• To calculate the area under the curve at a point

• To find the slope of the tangent line at a point

• To check if a function is increasing or decreasing at a point

• To determine if a function is continuous at a point

What does it mean if a function is differentiable at a point?

• The function is not defined at that point

• The function is decreasing at that point

• The function is continuous at that point

• The function has a maximum at that point

If a function is not continuous at a point, what can be said about its differentiability?

• It might be differentiable

• It is definitely differentiable

• It is undefined

• It is not differentiable

What happens to the slope of a function with a discontinuity as x approaches the point of discontinuity?

• The slope approaches infinity

• The slope approaches zero

• The slope becomes undefined

• The slope remains constant

What is a removable discontinuity?

• A point where the function can be redefined to be continuous

• A point where the function has a jump

• A point where the function has a vertical asymptote

• A point where the function is not defined

How does a removable discontinuity affect differentiability?

• It makes the function continuous

• It has no effect on differentiability

• It prevents the function from being differentiable

• It makes the function differentiable

Can a function be continuous at a point but not differentiable? Provide an example.

• Yes, for example, a linear function

• Yes, for example, the absolute value function

• No, all continuous functions are differentiable

• No, it cannot

Why is the absolute value function not differentiable at its vertex?

• It is not continuous at the vertex

• It has a sharp corner at the vertex

• It is not defined at the vertex

• It has a vertical tangent at the vertex

What is the significance of the slope of the tangent line in differentiability?

• It defines the area under the curve

• It shows the maximum value of the function

• It indicates the rate of change of the function

• It determines the continuity of the function

What can be concluded if a function approaches different values from the left and right at a point?

• The limit does not exist at that point

• The function has a maximum at that point

• The function is continuous at that point

• The function is differentiable at that point