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Q: Proving that $f(\bar Z)\subset\overline {f(Z)}$ when $f$ is a continuous map

SerphI'm trying to solve this question from my textbook: Let $f:X\rightarrow Y$ be a continuous map and let $Z \subset X$. Prove the inclusion $f(\bar Z)\subset\overline {f(Z)}$. Thanks in advance for any help!

as a duplicate of
Q: Continuity and Closure

Holdsworth88As a part of self study, I am trying to prove the following statement: Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where $\overline{A}$ denotes the closure of an arbitrary set $A$....

However, the later is about both implications, and it is more focused on the implication not asked int the first one.
Do we have a better choice for a duplicate?

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