# Accumulation points of the domain of a function

demonelite123
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?

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.