Can someone help me to show that Q is not homeomorphic to N.

We know that there exists a bijection between Q and N. How can I show that f:Q onto N is not continuous ?

I also have trouble with showing any isometry is continuous and is a homeomorphism.

Part 1)

We know that f: M onto N is an isometry if d_N (fp, fq) = d_M (p,q) for each p,q in M. So, f preserves distance. We now want to show that f is continuous. Assume that f is not continuous, then there exists x in M such that f is not continuous at x. This implies that there exists epsilon greater than 0 st for each delta greater than 0 and for each y in M, we have d_M (x,y) less than delta implies d_N (f(x), f(y)) greater than or equal to 0.

I am not able to see a contradiction here. Anyone can help pls?

Part 2)

We want to show that f is a bijection, but this is given in the definition of isometry. We need to show that f is continuous, this will be given if I can prove part 1). The last proof is to prove that inverse of f is continuous. Anyone can help?