let $f$ be a entire function and $B$ is bounded open set. $\partial(f(B)) \subset f(\partial B)$.

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let $f$ be a entire function and $B$ is bounded open set.

Then is it true that

$\partial(f(B)) \subset f(\partial B)$.

Hint is given that use open mapping theorem . But i got stuck how to use it. please provide some hint