# Accumulation points of the domain of a function

let a function f: A -> R^m where A is a subset of R^n. if a is an accumulation point of the domain of f, the set A, then f(A) is an accumulation point of the set R^m.

i'm not sure if this statement is true or not but if it is, how would I prove this result? i was thinking of using the fact that if a function is continuous and the sequence x_n approaches x, then f(x_n) approaches f(x). but the function need not be continuous for a limit to exist.

the question i'm trying to answer for myself is that, if a is not an accumulation point of the domain of a function then the limit as x approaches a is not well defined since it could be any number you choose and you could prove the result vacuously. so if a is an accumulation point of the domain does that necessarily mean that f(a) is an accumulation point as well?

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jgens
Gold Member
How about we consider $f:\bbold{R} \to \bbold{R}$ defined by $f(x) = 0$? Take an accumulation point in the domain of $f$ and see if you find an accumulation point in the image set.

ah i see. in this case since 0 is the only possible point in the codomain of f(x), f(A) is not always an accumulation point when A is an accumulation point.

are there cases where having f(A) be an accumulation point would be useful? as in proving some theorem or leading to some other concepts? or does having an accumulation point at f(A), given A is an accumulation point, not have much further significance whatsoever?