Determine the points of discontinuity. State the type of...

Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left-or right-continuous.
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f(x)=\lfloor x\rfloor
$$

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Key Concepts

Continuity

Continuity refers to the property of a function where small changes in the input lead to small changes in the output, without sudden jumps or breaks. A function is said to be continuous at a point if the limit as the input approaches that point exists and equals the function's value at that point.

Discontinuity

A discontinuity is a point in the domain of a function where the function is not continuous. This means there is a break, gap, or jump in the graph of the function at that point, and the usual limit conditions for continuity fail.

Types of Discontinuities

Discontinuities can be categorized into several types, including removable discontinuities (where a 'hole' exists in the graph that can be 'filled' by appropriately defining the function value), jump discontinuities (where the left-hand and right-hand limits exist but are not equal), and infinite discontinuities (where the function approaches infinity near the discontinuity).

One-sided Limits and Continuity

One-sided limits analyze the behavior of a function as it approaches a specific point from the left or the right. This is important for determining whether a function is left or right continuous at a given point, meaning the function's limit from one side equals the function's value at that point.

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