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The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable?

Something to the effect of:

##A \sim B\text{ and let } f \text{ be a continuous function X} \to \text{Y:}##

##\text{by definition, if } f \text{ is continuous, then there exists an } \epsilon \text{ for every } \delta \text{ such that ... blah...so }##

##f(A) \sim f(A+\epsilon) \sim f(A+n\epsilon) \text{ and by induction } f(A) \sim f(B) \text{ for large enough n}##

is that strong enough? Since A+n*epsilon = B it goes to f(B). Would that be necessary in the proof?

How can I do a proof with composite continuity?