Exercise 18.2

Question

Suppose that $f: X 	o Y$ is continuous.
If $x$ is a limit point of the subset $A$ of $X$, is it necessarily true that $f(x)$ is a limit point of $f(A)$?

Dan Whitman · Accepted Answer

This is not necessarily true.
  \begin{proof}
    As a counterexample consider a constant function $f: X 	o Y$ defined by $f(x) = y_0$ for any $x \in X$ and some $y_0 \in Y$.
    It was shown in Theorem~18.2 part~(a) that this is continuous.
    However, clearly $f(A) = \left\{y_0ight\}$ for any subset $A$ of $X$.
    So even if $x$ is a limit point of $A$, no neighborhood of $f(x)$ can intersect $f(A)$ in a point other than $f(x) = y_0$ since $y_0$ is the only point in $f(A)$!
    Therefore $f(x)$ is not a limit point of $f(A)$.
  \end{proof}

Exercise 18.2

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Exercise 18.2

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