Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (0, 3) at which f is not continuous.

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Identify the definition of continuity: A function f(x) is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit of f(x) as x approaches c is equal to f(c).

Examine the graph of the function f in the interval (0, 3) to identify any points where the function might not be continuous. Look for any breaks, jumps, or holes in the graph.

For each point of interest identified in the previous step, check if f(x) is defined at that point. If f(x) is not defined, the function is not continuous at that point.

Next, check if the limit of f(x) as x approaches the point from both the left and the right exists and is equal. If the left-hand limit and the right-hand limit are not equal, the function is not continuous at that point.

Finally, verify if the limit of f(x) as x approaches the point is equal to f(c). If any of these conditions are not met, the function is not continuous at that point. List all such points in the interval (0, 3).

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주요 개념

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Continuity of a Function

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 of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.

Differentiability of a Function

A function is differentiable at a point if it has a defined derivative at that point, which implies that the function is smooth and has no sharp corners or vertical tangents. Importantly, a function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability.

Identifying Discontinuities

Discontinuities in a function can occur in several forms, including removable, jump, and infinite discontinuities. To find where a function is not continuous, one must analyze the graph for any points where the function does not meet the criteria for continuity, such as breaks in the graph or undefined values. This analysis is crucial for determining the intervals of continuity and differentiability.

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>a. Find the values of x in (0, 3) at which f is not continuous.