Prove the image of a connected set is connected.

Problem: Let f:R^n -> R^m be continuous. Suppose A is a connected subset of R^n. Prove that f(A) is connected.

My proof:

Now f(A) = {f(x) : x in A}, and assume f to be an increasing function.

Now assume to the contrary that f(A) is not connected.

Let the two points d and e be in the set of f(A), then there exists open sets U and V in R^n such that they separate d and e.

Now d = f(a) and e = f(b) for a and b in A. Now assume a < b.

Since A is connected, A has Intermedian Value Property, that is, for every continuous function f, there exists a point z in A such that f(a) <= f(z) <= f(b) by the Intermedian Value Theorem.

But that contradict our assumption as it would be impossible for U and V to separate f(A).

Therefore we conclude f(A) is connected.

Q.E.D.

Yeah, I know this one is not really proper... But this is the best I could sequeeze out after almost two days of thinking...

Thanks, please check.

KK