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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?