= Stability of a supremum under uniform perturbations
{title2=$|\sup_X f-\sup_X g|\leq\|f-g\|_\infty$}
Suppose $f,g:X\to\mathbb R$ are <bounded functions> on the same nonempty <set>, and the <supremum norm> of their difference is at most $\delta$. Then their optimal values differ by at most the same amount:
$$
\boxed{\left|\sup_{x\in X}f(x)-\sup_{x\in X}g(x)\right|\leq\|f-g\|_\infty\leq\delta.}
$$
Indeed, $f(x)\leq g(x)+\delta\leq\sup_X g+\delta$ for every $x$, so $\sup_X f\leq\sup_X g+\delta$; exchange $f$ and $g$ for the other direction. Adding a constant shifts a <supremum> by that constant, a basic compatibility between <upper bounds> and <addition>.<Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/1.2 The set of real numbers>[Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl; §1.2.3, Proposition 1.2.6(i): translating a nonempty bounded-above set translates its supremum]{ref}
For a chosen example on $[-1,1]$, take $f(x)=-x^2$ and $g(x)=-x^2+0.2x$. Their <supremum error> is $0.2$, since $|g(x)-f(x)|=0.2|x|$. <Completing the square> gives
$$
g(x)=-(x-0.1)^2+0.01.
$$
Thus $f$ has <maximum> zero at $x=0$, while $g$ has <maximum> $0.01$ at $x=0.1$. The change in optimal value, $0.01$, is smaller than the certified bound $0.2$. For the vertical translation $h(x)=f(x)+0.2$, the bound is exact: its <supremum error> and optimal-value change are both $0.2$.
\Image[/supremum-uniform-perturbations-and-maximizers.png]
{title=Uniform error controls the optimal value; maximizing inputs require additional control}
{description=Original <polynomial> examples on $[-1,1]$. Left: $-x^2$ and $-x^2+0.2x$ lie within a vertical error band of $0.2$, while their <maxima> differ by $0.01$. Right: the small tilts $\varepsilon x$ and $-\varepsilon x$ have maximizing inputs $1$ and $-1$ for every $\varepsilon>0$; solid lines use $\varepsilon=0.1$ and dashed lines use $0.02$. These chosen examples are not measured errors or claims about an arbitrary optimizer.}
A stable optimal value does not guarantee a stable maximizing input. On $[-1,1]$, the two <linear functions> $f_\varepsilon(x)=\varepsilon x$ and $g_\varepsilon(x)=-\varepsilon x$ satisfy
$$
\|f_\varepsilon-g_\varepsilon\|_\infty=2\varepsilon\longrightarrow0,
\qquad\sup f_\varepsilon=\sup g_\varepsilon=\varepsilon.
$$
For every positive $\varepsilon$, their unique maximizing inputs remain two units apart, at $1$ and $-1$. Both <functions> approach the flat zero <function>, which has many maximizing inputs. Consequently an accurate <objective function> value is not by itself a certificate that the optimizing decision is accurate.
The estimate also gives a convenient consequence of <uniform convergence>. If $f_n$ converges uniformly to a <bounded function> $f$ on a nonempty $X$, then $\sup_X f_n\to\sup_X f$ for the bounded tail of the <sequence>. Uniform control handles the changing input that defines the best value. <Pointwise convergence> alone supplies no such control; the existing examples under <uniform convergence> illustrate that distinction.
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