Currently, we are examining when a function suddenly takes on a value which is a drastic change from all of the function values around it. Or, when a function just drops a value at a single domain number.

We have expectations based on the surrounding function values and then these expectations are not met.

In order to arrive at these expectations, we need some function values for domain numbers “around” the single domain number under examination.

$\blacktriangleright$ We need some filled space inside the domain around our domain number.

$\blacktriangleright$ We need some wiggle room inside the domain around our domain number.

We need space to gather up function values from which we are drawing our expectations. We need this room to provide a cushion around the domain number where we can see what the function is doing. Where we can observe its behavoir.

We need this space to be there no matter how close our examination is.

Our close space must work for all levels of inspection.

We are just beginning this investigation, so we want to avoid crazy behavior. Let’s just start with a nice story.

Learning Outcomes

In this section, students will

• contemplate space resulting from open intervals.