Question related to continuity in Topology ( A proof i am unable to understand) I am self studying Topology from C.Wayne patty and I am unable to think about a step in proof of a theorem in Section 1.6 .
Adding image ->
Note that i have question in only (c) ->(d) part of the proof.

In that part I am unable to understand how in 2-3rd line assuming x to be limit point of $f^{-1}(B)$ implies f(x) $\epsilon$ closure of $ f(f^{-1}(B)) $ .

I tried using definationof a limit point but it's not clear to me.
Please help.

That is the step that actually uses the hypothesis (c). In (c) take $A$ to be $f^{-1}[B]$; we’re assuming that $p\in\operatorname{cl}A$, so by (c) we have $f(p)\in\operatorname{cl}f[A]=\operatorname{cl}f\big[f^{-1}[B]\big]$.
