Math 468

Problem set 4, due Friday, February 13

1. Let f : R -> R be the function defined by f(x)=x². Show that f is continuous, using only the definition of continuity.

2. Recall that a topological space X is connected if the only subsets of X that are both open and closed are X and the empty set, ø. Given a function f:X -> Y, let f(X) denote the image of f, i.e., all the points z of Y which are of the form z=f(u) for some u in X. Show that if X is connected, then f(X) is connected.

3. A subset C of Rⁿ is bounded if for some number r, C is completely contained within the ball of radius r about the origin.

(a) Must the image of a bounded set be bounded? Why not? (I.e., give a counterexample.) What about preimages?

(b) Must the image of an unbounded set be unbounded? Why not? (I.e., give a counterexample.) What about preimages?

(c) Must the image of a closed and bounded subset of Rⁿ be closed and bounded? This one is harder; rather than trying to answer definitively one way or the other, work out some examples. (Take some familiar functions from R to itself, or from R² to R, or whatever, and see what the images of some closed and bounded sets are.)

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