For a ringed topological space$(X,O_X)$,let $f:X\to Y \space $be a continuous map between topological spaces. We know that $f_*O_X$ defines a sheaf on Y. So is the stalk of $f_*O_X$ just the same as the stalk of $O_X$, i.e.,
${(f_*O_X)}_{f(x)}={(O_X)}_{x}$
Does this hold?
We can explicitly write out the stalks in terms of their definition (as colimits):
$$\mathcal{O}_P := \varinjlim_{U \ni P} \mathcal{O}(U)$$ $$\mathcal{f_{*}O}_P := \varinjlim_{V \ni f(P)} (f_{*}\mathcal{O})(V) := \varinjlim_{f^{-1}(V) \ni P} \mathcal{O}(f^{-1}(V))$$
As we can see with this computation, and using Lazzaro Campeotti's counter-example, the stalks need not coincide.