Understanding Discontinuities in Mathematics | Types and Meanings

discontinuities go into the chart like cns, but discontinuities they are NOT _______ since they make the function UNDEFINED

Discontinuities, also known as points of discontinuity, are specific values of the independent variable (usually denoted as x) in a function where the function is undefined or does not have a well-defined value

Discontinuities, also known as points of discontinuity, are specific values of the independent variable (usually denoted as x) in a function where the function is undefined or does not have a well-defined value. In other words, these are values for which the function “breaks” or exhibits some sort of “gap” or “jump”.

Discontinuities are NOT holes in a graph, since holes refer to missing points in a function where there is a well-defined limit. Instead, discontinuities occur when the function cannot be evaluated at certain points or when the function does not approach a single value as the independent variable approaches a particular value.

The three main types of discontinuities are:

1. Removable Discontinuity: These occur at points where the function has a hole in the graph. The function is undefined at that point, but the limit of the function as x approaches that point exists and can be filled in to make the function continuous.

2. Jump Discontinuity: These occur at points where the function “jumps” from one value to another without any intermediate values. The left-hand and right-hand limits exist, but they are not equal.

3. Infinite Discontinuity: These occur at points where the function approaches positive or negative infinity. The left-hand or right-hand limits approach infinity, implying that the function grows without bound as x approaches the discontinuity.

It is important to note that when a function has a discontinuity at a certain point, it does not mean that the function is entirely meaningless. It simply indicates that the function is undefined or behaves in a non-continuous manner at that specific point.

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