Is there a way to have Plot draw points on a function where the function is undefined as circles as in textbooks?

Question

A very common feature of graphs of functions used throughout textbooks are simple indications, such as open circles, of the points at which a function is undefined. Is there a way to do this in Mathematica?

I understand that I can use Epilog (or Prolog, or other explicit drawing graphics functions) to "manually" draw these elements, but us there a way to automatically do this, or at least return the coordinates of such points for use as arguments to Epilog?

Answer 1

By using Exclusions and ExclusionsStyle like this

Plot[1/Sin[π x], {x, -3, 3},
  Exclusions -> Sin[π x] == 0, 
  ExclusionsStyle -> {Thin, Directive[Red, PointSize[0.02]]}
]

you can define the location of the discontinuities and the style for poles and discontinuity boundaries, respectively.

Also the Wolfram Blog, March 2011 has more examples on this topic.

Answer 2

You can use the ExclusionsStyle option to control how Mathematica renders regions excluded from the Plot.

Edit Doing something like ExclusionsStyle -> {None, {{Black, PointSize[Large]}}} will put a large point at the excluded points. For example, modifying a function from the help documentation: