We were talking about the intermediate value theorem (can't get from here to there without sometime being in between them) in my real analysis class and the thought occurred to me that the interval is a very restrictive notion. How does this generalise to more dimensions?
Is there a natural way to extend the IVT beyond real valued functions? I could see that the length/radius of a set of points in R^2 or C would have this property (since we have length in R, and the curve is continuous implies the changes in its distance are continuous), but I don't see what it means for say the helix f:R-->R^3 where f(t)=(cos(t), sin(t), t) except when considered as three separate continuous functions. (i.e., let a = 0, b = 2*pi, and see that nothing more can be said than that there exist z values between 0 and pi for which the z component of f(t) obeys the intermediate value property).
Is the problem in the notion of an interval, or of betweenness? Is there a better way to describe this phenomenon when talking about parameterised curves?
1 comment:
So the answer is either consider each continuous function as a continuous component and go from there, or use the other map from R^3->R, arc length. Then all it says is that you can't have a continuous curve of non-zero length without having to have had all the lesser length curves along the way.
A useful note, for the intermediate value thereom, chasing down a preimage c of a point d interior to the interval of the image of the endpoints a and b is equivalent to finding a supremum for the set of deltas with epsilon equal to d-f(a); or finding a supremum of the delta set where epsilon is f(b)-d.
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