Determine whether the given function f is continuous on the indicated intervals. (a) [-1,3] (b)

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Determine whether the given function f is continuous on the...

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition of continuity provides a formal framework to assert that a function is continuous. It requires that for any given tolerance (epsilon) in the function’s output, there exists a corresponding range (delta) around the input such that any change within that range results in an output change smaller than the specified tolerance. This approach underpins the rigorous analysis of continuous behavior on intervals.

One-Sided Limits

One-sided limits are crucial for checking continuity at the boundary points of an interval. For functions defined on closed or half-open intervals, you approach an endpoint only from the side that lies within the interval. The continuity at the boundary relies on the limit from the appropriate direction equaling the function's value at that endpoint.

Continuity

Continuity of a function means that its graph can be drawn without lifting the pencil. More formally, a function is continuous at a point if the limit of the function as the input approaches the point equals the function's value at that point, ensuring there are no jumps, holes, or vertical asymptotes in the graph.

Closed and Half-Open Intervals

When discussing the continuity of a function on an interval, it is important to consider the nature of the interval. For closed intervals, continuity must be verified at every point including the endpoints; at the endpoints, one uses one-sided limits to check continuity. For half-open intervals, the function must be continuous on the interior and at the included endpoint, while the open endpoint is not required to satisfy the continuity condition.

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