Proof
(related to Proposition: Continuity of Compositions of Functions)
 By hypothesis, $(X,\mathcal O_X),$ $(Y,\mathcal O_Y),$ $(Z,\mathcal O_Z)$ are topological spaces, and $f:X\to Y$ and $g:Y\to Z$ are continuous functions.
 By definition, the inverse image under $g$ of an open set in $Z$ is an open set in $Y.$
 Again, the inverse image under $f$ of this open set in $Y$ is an open set in $X.$
 Thus, the composition $g\circ f:X\to Z$ is continuous.
∎
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984