Function between two metric spaces?

Question

I need to come up with:

• two metric spaces ( X , d ) and ( Y , p )

• A continuous function f: X → Y

• A Cauchy sequence {x_n} in X that isn't mapped to a Cauchy sequence in Y

My idea was to make the first metric space ℝ with the "usual" euclidean metric, and the second metric space would be ℝ with the discrete metric. Then the function would be anything continuous (like y=x or y=x^2). Since the second space has elements that are always a distance of 1 away from each other, you'd never map a Cauchy to a Cauchy...

But I'm awful at real analysis so I don't doubt there is an error here.

Answer 1

Hint: Let $X$ be the reals in $(0,1)$ with the usual metric, and $Y$ the reals with the usual metric. Let $f(x)=\tan(\pi x/2)$.