We have one slight hiccup in our open interval version of closeness and that is endpoints of intervals.

Graph of \(y = K(r)\).

If we want to examine the domain number \(2\) and its weird function value \(-3\), then our open interval idea is in trouble. There is no filled space to the right of \(2\) in the domain where we could carve out an open interval.

Actually, we don’t need it.

We just need the domain numbers that are close to \(2\).

We need all of the domain numbers that are close to our domain number under observation.

We don’t need all of the real numbers that are close to our domain number under observation. We just need the real numbers that are actually members of the domain.

Slight Modification

Remember, we are after some language that works ALL of the time, even for endpoints.

“Space” will refer to domain space that exists “around” our domain number under observation.

\(\blacktriangleright \) Language:

\(\blacktriangleright \) Notation:

The intersection picks out the numbers that are both in the \(\delta \)-interval and in the domain.

\(\blacktriangleright \) This actually fixes a hidden problem. If our expectations are disrupted by a missing domain number, then we shouldn’t be including that number in our \(\delta \)-interval. However, if we stipulate that we are only looking at domain numbers inside the \(\delta \)-interval, then everything works.

Tools

We have our communication tools.

• Open intervals for space.
• \(\delta \)-intervals for closeness

We are ready to begin our algebraic description of interrupted expectations in function values.