= Bijective monomial cusp parametrization with nonregular inverse
The <morphism> $\mathbb A^1\to V(x^4-y^3)$ given by $t\mapsto(t^3,t^4)$ is <bijective> on <points> over an <algebraically closed field>, but is not an <isomorphism>. At a nonzero curve <point>, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on <coordinate rings> has <image> $k[t^3,t^4]\subset k[t]$, which lacks the <monomial> $t$: all its <monomial> exponents have the form $3a+4b$, with $a,b\geq0$. Nor can $t$ be regular at the <cusp>, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\neq0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus <pointwise> bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one <singular point>.
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