You can use the inequality and write .
Let be a metric space. Let be a continuous function. Let be defined by . Show that is continuous.
So what we know is that is continuous, so for any , there exists such that:
But what we want to do for any is to produce such that:
I have a feeling we'll want to produce in terms of . I was thinking breaking something up using a triangle inequality, but I've tried a few different ways that have all led to dead ends.