In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?
This was not quite what I was thinking about. But I think it helps clarify the question I had. I have not thought much about the topology of the hyperreals but what I'm asking is: