help needed with connected subsets, topology please?

Question

Exercise 4. Let I be an open interval in R and let f : I -> R be a differentiable function.
(1) Prove that the set T = {(x, y) in I × I : x < y} is a connected subset of R2 with the standard
topology.
(2) Let g : T -> R be the function defined by
g(x, y) =(f(x) − f(y))/(x − y)
.
Prove that g(T) (open) is a subset of f '(I) is a subset of g(T) (closed).

Any help on this would be much appreciated. Thanks

? · Accepted Answer

If f is differentiable, then it is a smooth function on I.  Geometrically speaking, what you're doing here is drawing this differentiable function on the plane, and f has no holes.  (Remember when you need to calculate volumes of surfaces of revolution?).  All you need to show is that being differentiable on I means there are no points of disconnectedness.

For part (2), recall your definition of derivative.  Think of it as the convergence of a sequence of secants to the unique tangent.  Recall that a set is closed if it contains all of its limit points.

I think with these hints and some effort on your part, you should be able to get the answer.

help needed with connected subsets, topology please?

1 Answer