Return to Revisions

Do you want an example where you can give the proof, not just the statement, in the lecture too? Lots of other suggestions are great uses of connectedness, but in one lecture where you first introduce the concept I don't think they can be fully explained. Here is one simple result which I think can: a function on an interval I which is locally a polynomial is globally a polynomial. That if, if f : I ---> R and around each a in I there's a neighborhood and a polynomial p_a(x) such that f(x) = p_a(x) for all x in a, then there is a single polynomial p(x) such that f(x) = p(x) for all x in I. The point is that polynomials that agree at infinitely many points are equal everywhere and connectedness lets you show a polynomial locally equal to f near one point has to be locally equal to f everywhere. Admittedly this is not an important result compared to the other suggestions, but it illustrates in a nice way one of the points of connectedness: how it turns local information into global information.