= Solution
Let $X=\bigcup_{i=1}^rU_i$ be the stated finite affine cover. By the description of sections on a <principal open subscheme>, after increasing denominators separately we may write
$$
b|_{(U_i)_f}=a_i/f^{n_i}
$$
for some $a_i\in\Gamma(U_i,\mathcal O_X)$. Multiplying the numerators by powers of $f$ lets us use one exponent $n$ for every $i$.
On $W_{ij}=U_i\cap U_j$, the section $a_i-a_j$ vanishes after restriction to $(W_{ij})_f$. Each $W_{ij}$ is <quasi-compact>, so part (b) supplies $m_{ij}$ with $f^{m_{ij}}(a_i-a_j)=0$ on $W_{ij}$. Choose one $m$ valid for all the finitely many pairs. The sections $f^ma_i$ now agree on every overlap, and the gluing axiom of a <sheaf of rings> produces $a\in\Gamma(X,\mathcal O_X)$ with $a|_{U_i}=f^ma_i$. On $X_f$,
$$
a=f^{m+n}b.
$$
Thus \b[some power of $f$ times $b$ extends to a global <regular function> on $X$].
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