- #1
Silviu
- 624
- 11
Homework Statement
Find ##f:R \to X##, f-continuous, where X is the discrete space.
Homework Equations
The Attempt at a Solution
f is continuous at p if for any ##\epsilon > 0## there is ##\delta >0## such that ##d(f(x),f(p))<\epsilon## for all x such that ##d(x,p)<\delta##. Let ##\epsilon = 1##. As X is the discrete space, only f(p) satisfies ##d(f(x),f(p))<\epsilon##. So for a whole open interval (which becomes closed due to continuity) we have ##f([p-\delta,p+\delta])=f(p)##. From here I can see that only constant functions would work here, but I am not sure how to continue. I was thinking to do the same reasoning at ##f(p+\delta)##, and extend this over the whole R, but if the intervals get smaller and smaller this might not work. How should I continue?