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It's not quite clear what you mean by "mathematical investigations", but I think that this is basically what hundreds (more?) of us use the Sage notebook for on a daily basis.

You can go to the main Sage notebook server at the University of Washington and try it out (there are a number of others). I especially recommend going to the published worksheet link and using the search box for various topics you may find useful.

The only catch is that anyone using the Sage notebook (currently) has to log in and actually view a page. We will probably soon have the potential for "live" interactive things that do not require login, or even download of a plugin (so, easier to use than Mma Demonstrations), but that is still working through the review process, I believe.

If this is not quite what you were asking, please just update your question. I have personally used Sage in the classroom to do exactly this - both live calculations and investigations - for over four years, and am very satisfied with the results. There are other educators who discuss these kinds of meta-issues at the Google group for education and Sage, which you may wish to join and participate in; the group for support can handle most technical questions.

Good luck!