The sentence of the title is written in J.L. Kelley's book General Topology, page 261 (at the appendix) and said:
It is to be noticed that if $X$ is a subset of the domain of $f$ then $f(X)$ is not
$$\{y:\mbox{for some } x, x\in X \mbox{ and } y=f(x)\}, \tag{1}$$
but I'm not sure about the reason. I suppose it is related to the definition of $f(x)$:
$$ f(x)=\bigcap\{y:(x,y)\in f\}. \tag{2} $$
My unique idea is that (1) needs to be change to
$$ \{y:\mbox{for some } x, x\in X \mbox{ and } (x,y)\in f\}. $$
Is right my ''solution''?
PD: I must confess I'm quite confussed about $f(x)$ notation. Because sometimes I read $f(x)=y$ was a notation to know $(x,y)\in $, as long as $x\mathcal R y$ to $(x,y)\in \mathcal R$. Is that possible from a rigorous point of view?
While $f(A)$ is a very common abuse of notation, Kelley uses the notation $f[A]$ for the image set of $A$. See p.11.