On approximate antigradients

Abstract:

For $n \in \mathbb {N}$ and $I = [0,1]$, let ${I^n}$ be the unit cube and ${\lambda ^n}$ the Lebesgue measure in ${\mathbb {R}^n}$. It is proved that if $f:{I^n} \to {\mathbb {R}^n}$ and ${F_0}:{I^n} \to \mathbb {R}$ are continuous and $\varepsilon > 0$, then there exist a continuous $F:{I^n} \to \mathbb {R}$ and an open set $W \subset {({I^n})^ \circ }$ with ${\lambda ^n}(W) = 1$ such that (i) $\nabla F$ exists and is continuous on $W$, (ii) $||\nabla F(x) - f(x)|| < \varepsilon \;\forall x \in W$, and (iii) $|F(x) - {F_0}(x)| < \varepsilon \;\forall x \in {I^n}$, where $||y|| = {\left ( {\sum \nolimits _{j = 1}^n {y_j^2} } \right )^{1/2}}\;\forall y \in {\mathbb {R}^n}$.