Questions regarding properties

[MathewsMD]

If there is a graph that has a discontinuity (e.g. at x = 3) and then the there is a point much lower than the limit at 3, is that point at 3 considered a local minimum? Also, if the graph starts w/ a discontinuity at its "highest point", does it have an absolute maximum?

 

[haruspex]

I'm guessing you mean, in the first question, that the function would have been continuous at x if f(x) had been some value y, but the discontinuity is that it has a lower value than y. If so, yes, I would say that constitutes a local minimum.
For the second question, I guess you mean, again, that the discontinuity consists of the function taking a lower value than the approach limit, and nowhere takes a value as high as the approach limit. If so, there is no absolute maximum.