Singularity

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holes

A singularity is the second kind of interruption in our expectations.

The word “singularity” covers a lot of weird situations. We are just at the beginning of the singularity story. For us, singularity will be just like a discontinuity, except the number under examination is not a domain number. So, the function doesn’t have a value there.

Jump Singularity

A jump singularity occurs when the function is continuous to the left and right side of the singularity, but the two sides do not match up.

Removeable Singularity

A removeable singularity occurs when the function is continuous to the left and right side of the discontinuity, and the two sides do match up.

This type of singularity can be ”removed” simply by defining the function value at that one number to plug the hole.

Asymptotic Singularity

The third type of singularity is when the values of the function become unbounded near a real number. Graphically, these are represented with asymptotes. There are several possible configurations.

Singularities where both sides become unbounded.

Singularity where only one side becomes unbounded.

Those are the three main types of singularities we will encounter. Rational functions naturally have asymptotic and removeable singularities. Other than that, we use piecewise defined functions to create singularities.

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More Examples of Continuity