Sketch the graph of a function f that is continuous except for the stated discontinuity. Removab

step 1 So we have to just have a graph and I'm just looking at the different pieces that the moment I s

Sketch the graph of a function f that is continuous except...

Key Concepts

Jump Discontinuity

A jump discontinuity happens when the left-hand and right-hand limits exist at a point, but they are not equal. This results in a sudden 'jump' in the graph, where the function abruptly changes value, making it impossible to 'connect' the two sides of the discontinuity.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They are commonly used to create functions with specific behaviors, such as introducing discontinuities, where each piece can be separately defined to account for conditions like removable or jump discontinuities.

Continuity

Continuity of a function implies that the graph can be drawn without lifting the pencil from the paper. A function is continuous at a point if the limit as you approach that point is equal to the function’s value at that point. This concept is central to understanding the behavior of functions on intervals where they show no breaks or jumps.

Removable Discontinuity

A removable discontinuity occurs at a point where the limit of the function exists, but the function itself is not defined or does not take that limit value at that point. This often appears as a ‘hole’ in the graph, which can sometimes be 'fixed' by appropriately defining the function value at that point.

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