= Solution
For an <affine variety> $V\subseteq\mathbb A_k^N$, one construction defines $\mathcal O_V(U)$ as the functions $U\to k$ which locally have the form $p/q$, where $p,q$ are elements of the <coordinate ring> $k[V]$ and $q$ does not vanish on the neighbourhood in question. Restrictions are restrictions of functions, and locality gives the <sheaf> axioms. In particular
$$
\boxed{\mathcal O_V(D(f))=k[V]_f,\qquad(\mathcal O_V)_x=k[V]_{\mathfrak m_x}.}
$$
The second formula identifies its <local rings>.
In the classical convention, an <algebraic variety> over $k$ is an irreducible <ringed space> with a <sheaf> of $k$-valued functions, admitting a finite <open cover> by spaces isomorphic to affine varieties, and satisfying the <separated variety> condition that its diagonal is closed in $X\times X$. Its topology is Noetherian. One can instead allow reducible reduced varieties; the arguments below still work with the denominator-clearing version of the <localization> argument. A <morphism of varieties> is a continuous map whose pullback takes local <regular functions> to <regular functions>, equivalently a morphism of locally ringed $k$-spaces. The maps on <local rings> are local because a function nonzero at the image point stays nonzero at the source point.
An <affine variety> supplies its own finite affine cover, is Noetherian, and is separated: its diagonal in $V\times V$ is cut out by the differences of corresponding coordinates. A <regular map> between affine varieties pulls coordinate functions back to <regular functions>. It is continuous because the inverse image of any <polynomial> zero set is the zero set of the pulled-back <regular functions>; on affine charts such zero sets are closed. Pulling back a locally represented fraction $p/q$ gives a regular fraction wherever its denominator is nonzero. Hence a regular map is a <morphism of varieties> in this definition.
For a morphism $\phi:X\to Y$ which is an <isomorphism> over each member $U_i$ of an <open cover> of $Y$, every fibre contains exactly one point. Thus $\phi$ is bijective. Its inverse is continuous and regular on every $U_i$, since there it is the given inverse of the local <isomorphism>. These inverses agree on overlaps, being inverses of the same map. They glue to a global inverse <morphism of varieties>. Therefore \b[being an <isomorphism> is local on the target].
Let $A=\Gamma(X,\mathcal O_X)$, and let $Y$ be affine with <coordinate ring> $B$. Given a $k$-algebra homomorphism $\alpha:B\to A$, choose a presentation $B=k[y_1,\ldots,y_N]/I$. The map
$$
\boxed{x\longmapsto\bigl(\alpha(y_1)(x),\ldots,\alpha(y_N)(x)\bigr)}
$$
lands in $Y$ because all the equations in $I$ become zero functions. Its coordinates are <global regular functions>, so it is a morphism on every affine chart of $X$, and therefore globally. On a target neighbourhood where a fraction $b/c$ is defined its pullback is $\alpha(b)/\alpha(c)$, proving that the induced map on <global sections> is exactly $\alpha$. This also proves independence of the chosen generators and uniqueness. This is the <affine-target adjunction for varieties>.
For $f\in A$, the natural map $A_f\to\Gamma(X_f,\mathcal O_X)$ sends $a/f^N$ to that <regular function>. In the irreducible convention, if $f\ne0$, $X_f$ is dense, so this map is injective: a global <regular function> vanishing there vanishes everywhere. Cover $X$ by finitely many affine charts $U_j$. A section $s$ on $X_f$ restricts on $U_j\cap X_f=D_{U_j}(f|_{U_j})$ to an element of $\Gamma(U_j,\mathcal O_X)_{f|_{U_j}}$. A common power $f^N$ clears all these finitely many denominators. The resulting regular sections on $U_j$ agree on the dense principal open in each overlap, hence agree there altogether and glue to a <global section> $a$. Thus $s=a/f^N$. If $f=0$, both sides are the zero ring of sections on the empty open. Consequently
$$
\boxed{\Gamma(X_f,\mathcal O_X)\cong A_f.}
$$
This is <localization of global sections on a principal open>. For reduced reducible varieties, equality on a principal open instead means that a sufficiently large power of $f$ annihilates the difference; finitely many charts and overlap refinements allow one common extra power. The same denominator-clearing proof then gives the stated <localization> <isomorphism> without a density assumption.
Now suppose $\sum_i g_if_i=1$ in $A$ and every $X_{f_i}$ is affine. Each $A_{f_i}=\Gamma(X_{f_i},\mathcal O_X)$ is a finitely generated $k$-algebra. Select finitely many generators, writing them as $a_{ij}/f_i^{N_{ij}}$. Let $B\subseteq A$ be the $k$-subalgebra generated by all $f_i$, all $g_i$, and all $a_{ij}$. It is finitely generated, and
$$
\boxed{B_{f_i}=A_{f_i}\quad\text{for every }i.}
$$
The inclusion from left to right is immediate, while the chosen generators give the reverse inclusion. This construction does not assume that $A$ was finitely generated in advance.
The reduced algebra $B$ is the <coordinate ring> of an affine variety $Y$ (irreducible when $X$ is). The map $B\hookrightarrow A$ produces $X\to Y$ by the preceding construction. The sets $D_Y(f_i)$ cover $Y$, since their functions generate the <unit ideal> already in $B$. Their inverse images are $X_{f_i}$, and on them the map is the <isomorphism> corresponding to $B_{f_i}=A_{f_i}$. The target-local argument above now proves
$$
\boxed{X\cong Y\quad\text{and in particular }X\text{ is affine}.}
$$
This is <affineness from a unit-ideal principal affine cover>. The printed sets are $X_{f_i}$; the missing index in the TeX aid is not a different hypothesis.
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