= Solution
For any <affine variety> and <point> $p$, with $\mathfrak m_p=\{f\in A(X):f(p)=0\}$, the standard <localization> description is
$$
\boxed{\mathcal O_{X,p}\cong A(X)_{\mathfrak m_p}.}
$$
Also $\mathcal O_X(D(g))\cong A(X)_g$ for a <principal open subset>. One way to see the latter is to realize $D(g)$ as the <affine variety> $\{(x,z):g(x)z=1\}$; its <coordinate ring> is $A(X)[z]/(gz-1)=A(X)_g$ and part (d) applies.
For this <cusp> use $x=t^3,y=t^2$, identifying $A(X)$ with $k[t^2,t^3]$ inside $k(t)$. The parametrization gives exactly the relation $x^2=y^3$; its kernel is generated by this binomial. For example reduce a <polynomial> modulo $x^2-y^3$ to $a(y)+xb(y)$; substitution has even powers from $a(t^2)$ and odd powers at least three from $t^3b(t^2)$, so it can vanish only when both <polynomials> are zero.
Write $g=x-y=t^2(t-1)$. The <open set> corresponds to $t\neq0,1$, and
$$
\mathcal O_X(U)=k[t^2,t^3]_{t^2(t-1)}.
$$
To simplify this <ring> without assuming the normalization coordinate is present, observe that in $A(X)$
$$
g^2=(x-y)^2=y(y^2-2x+y).
$$
Thus $y$ becomes a <unit> after inverting $g$, with inverse $(y^2-2x+y)/g^2$. Consequently $t=x/y$ belongs to the <localization>. Since $g=t^2(t-1)$ is a <unit>, both $t$ and $t-1$ become <units>. Conversely every original generator lies in the corresponding two-point <localization> of $k[t]$. Hence
$$
\boxed{\mathcal O_X(U)=k[t,t^{-1},(t-1)^{-1}].}
$$
Finally $p=(1,1)$ corresponds to $t=1$. It lies outside $U$, so compute its <local ring> from $X$, at $\mathfrak m=(x-1,y-1)$. The element $y$ is a <unit> there, hence $t=x/y$ belongs to $A(X)_{\mathfrak m}$. Every <polynomial> in $t$ nonzero at one is then a <unit>. Conversely a denominator from $A(X)\setminus\mathfrak m$ remains nonzero at $t=1$. These inclusions prove
$$
\boxed{\mathcal O_{X,p}=\bigl(k[x,y]/(x^2-y^3)\bigr)_{(x-1,y-1)}\cong k[t]_{(t-1)}.}
$$
This <smooth> <point> has the ordinary one-variable <local ring>, even though the global curve has a <cusp> at its origin.
The <localization> calculation describes <regular functions on a cusp with two points removed>.
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