Suppose … is continuous at …, say

Suppose $g(x)$ is continuous at $x=4$, say $g(4)=7$. Suppose $f(y)$ is continuous at $y=7$, say $f(7)=9$. Then $f(g(x))$ is continuous at $x=4$ and $f(g(4))=9$.
Proof $\epsilon$ is given. Because ______ is continuous, there is a $\delta$ such that $|f(g(x))-9|<\epsilon$ whenever $|g(x)-7|<\delta$. Then because ______ is continuous, there is a $\theta$ such that $|g(x)-7|<\delta$ whenever $|x-4|<\theta$. Conclusion: _If_ $|x-4|<\theta$ _then_ ______. This shows that $f(g(x))$ approaches $f(g(4))$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §4.1 The Chain Rule, #36