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 to identify points where the function is continuous. A function is continuous at a point if there is no break, jump, or hole at that point.
2. Check for differentiability at each point where the function is continuous. A function is differentiable at a point if it has a defined tangent line, meaning the graph is smooth and not sharp or vertical at that point.
3. Identify any points where the graph has sharp corners or cusps, as these are typically points where the function is not differentiable.
4. Look for vertical tangents or discontinuities, as these indicate points where the function is not differentiable.
5. Based on the graph, determine the points within the interval [-3, 3] where the function is both continuous and smooth, indicating differentiability.

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 well-defined. This requires the function to be continuous at that point and for the left-hand and right-hand limits of the derivative to exist and be equal. Points where the graph has sharp corners, vertical tangents, or discontinuities indicate non-differentiability.

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, it must first be continuous; however, continuity alone does not guarantee differentiability.

Closed Interval

A closed interval, denoted as [a, b], includes all 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 for the examination of its behavior at the endpoints and within the interval, ensuring that all points are considered when determining differentiability and continuity.