small edit needed(?) for continuity

[Alex-Jordan]

Theorem 1.5.12: Continuous Functions https://opentext.uleth.ca/apex-video/sec_continuity.html#thm_continuous_functions

says that certain functions are "continuous on their domains". For functions like tangent, the domain is neither an open interval nor a closed interval, which are the only types of set for which "continuous on a set" has been defined/discussed. Noting here to seek out a way to adjust the wording to make this fit well without becoming too verbose or technical.

[sean-fitzpatrick]

Can we just add something like:

<aside>
  <p>
    When the domain of a function <m>f</m> is a disjoint union of open intervals,
    we say that <m>f</m> is continuous on its domain if it is continuous on each interval.
  </p>
</aside>

Maybe there could be a second sentence with an xref to the example of the floor function to urge caution in the case of half-open intervals.

[APEXCalculus]

I like the basics of Sean's suggestion. How about relating it directly to the Theorem at hand, with the following adjustment:

For functions f (such as \tan x, \cot x, \sec x and \csc x) where the domain the function is a disjoint union of open intervals, we say that f is continuous on its domain if it is continuous on each interval.

[sean-fitzpatrick]

I added the following, next to the theorem stating which functions are continuous on their domains:

  <aside>
    <p>
      We have defined what it means for a function to be continuous on an interval,
      but many functions, such as <m>f(x)=\tan(x)</m>,
      have domains that are the union of more than one interval.
    </p>

    <p>
      If the domain of a function is a union of intervals, saying that a function is continuous on its domain
      means that the function is continuous on each of those intervals.
      But be careful to note that the converse is not true.
      As we learned in <xref ref="ex_contint2"/>,
      a function can be continuous on a collection of intervals, but not on their union.
    </p>
  </aside>

[sean-fitzpatrick]

I didn't include "disjoint" (unintentionally), or "open" (intentionally, because we can construct examples -- like square roots of polynomials -- where the domain is a disjoint union of closed intervals)

[sean-fitzpatrick]

Since Greg just merged changes in #168 I think I can safely close this.

[APEXCalculus]

Yes - I forgot that step. Thanks.

On Wed, May 11, 2022 at 11:01 AM Sean Fitzpatrick @.***> wrote:

Closed #156 https://github.com/APEXCalculus/APEXCalculusPTX/issues/156.

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