= Continuity bounds for the square function
{title2=$\delta=\min\{1,\varepsilon/(2|a|+1)\}$}
For $f(x)=x^2$ at any real $a$, <polynomial factorization> gives
$$
|f(x)-f(a)|=|x-a|\,|x+a|.
$$
If $|x-a|<1$, the <triangle inequality> yields $|x+a|\leq|x-a|+2|a|<2|a|+1$. Therefore the displayed choice of $\delta$ makes $|x-a|<\delta$ imply $|f(x)-f(a)|<\varepsilon$. This proves <continuity> and gives an explicit local error bound, rather than an inference from a drawing.
At $a=1$ and $\varepsilon=0.6$, this sufficient choice is $\delta=0.2$. The permitted <function inputs> satisfy $0.8<x<1.2$; their actual <function outputs> lie in $(0.64,1.44)$, inside the requested band $(0.4,1.6)$. A sufficient input window need not be the largest possible one.
\Image[/square-continuity-input-output-tolerances.png]
{title=An input window that guarantees the requested output tolerance for the square function}
{description=For the <polynomial function> $f(x)=x^2$ at $a=1$, the blue input window has $\delta=0.2$ and the orange output band has $\varepsilon=0.6$. The highlighted <function graph> segment lies inside their overlap. The illustration shows an exact mathematical example, rather than measured values.}
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