ac.commutative algebra - Filtration over tensor product

fa.practical evaluation – Is the pointwise supremum of a steady responsibility steady? retort

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fa.practical evaluation – Is the pointwise supremum of a steady responsibility steady?

Suppose $f(x , y)$ is steady in each variables. For any $epsilon > 0$ and a few $y_0$, let $h_{epsilon}(x) = max_{y^{‘}: | y^{‘} – y_0 | leq epsilon} f(x , y^{‘})$. It appears to me that $h_{epsilon}(x)$ is steady in $epsilon$ on $(0 , infty)$, that’s, for any $epsilon_n rightarrow epsilon_0 > 0$, one has $h_{epsilon_n}(x) rightarrow h_{epsilon_0}(x)$.

Is this correct? How ought to I show it?

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