Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


a. differentiable?


Give reasons for your answers.

Verified step by step guidance

1. Examine the graph of the function over the interval D: [-3, 3]. Look for points where the graph has sharp corners or cusps, as these are typically points where the function is not differentiable.
2. Identify any points where the graph has vertical tangents or discontinuities. These are also points where the function is not differentiable.
3. Check the endpoints of the interval, x = -3 and x = 3. Differentiability at endpoints requires one-sided derivatives to exist and be equal, which is often not the case.
4. Observe the smoothness of the graph between the points -3 and 3. If the graph is smooth and continuous without any sharp turns, the function is likely differentiable at those points.
5. Conclude that the function appears to be differentiable at points where the graph is smooth and continuous, excluding any sharp corners, cusps, or endpoints where differentiability is not guaranteed.

Key Concepts

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, meaning the function's graph has a tangent line that is not vertical. This requires the function to be smooth and continuous at that point, without any sharp corners or cusps. If a function is not differentiable at a point, it may be due to discontinuities or abrupt changes in direction.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. For a function to be differentiable at a point, it must first be continuous there; however, continuity alone does not guarantee differentiability.

Closed Interval

A closed interval, denoted as [a, b], includes all the numbers between a and b, including the endpoints a and b themselves. In the context of differentiability and continuity, analyzing a function over a closed interval allows us to evaluate its behavior at the endpoints and within the interval. This is crucial for determining where the function is differentiable or continuous.