Question On page 111, Number 2, we are asked, "If f:X to Y is continuous, if x is a limit point of a subset A of X is it necessarily true that f(x) is a limit point of f(A) ?"

I'm confused about the notation... isn't

f(x) an element of f(A)? f(A is a set, right? So even if f were a contant function so everything goes to 7, wouldn't f(x)=7 and f(A)= {7} I'm not sure if I should say it's the same or not.

Answer Yes, f(x) is an element of f(A). But that doesn't mean that f(x) is a limit point of f(A).

As for notation, f(A) = {f(x):x is in A}.

In fact, you've given an example where this fails. To be more precise: Let R be the set of real numbers, in the usual topology. Let f be the function from R to R such that f(x)=7 for all x in R.

Let A=R. Then 52 is a limit point of A, since every open set of R containing 52 also contains a point of A other than 52.

Now f(A)={7} and f(52)=7. So f(52) is not a limit point of f(A). The open interval (6,8), for example, is an open set containing f(52), but it does not contain any point of f(A) other than f(52).