The relationship between con't function and a compact set

[pantin]

suppose f:R^m -> R^n is a map such that for any compact set K in R^n, the preimage set f^(-1) (K)={x in R^m: f(x) in K} is compact, is f necessary continuous? justify.

The answer is no.
given a counterexample,

function f:R->R

f(x):= log/x/ if x is not equal to 0
f(x):= 0 if x=0

note, /x/ is the absolute value of x.


I don't quite get how to draw the image log/x/
and anyone can explain why ?

 

[morphism]

What exactly are you having trouble with?

 

[pantin]

let me try to plug in some number to the fn in the solution tomorrow...too late tonight, going to sleep.. thanks for asking :)