= Coordinate Lipschitz continuity
{title2=$|\partial_i f(x+te_i)-\partial_i f(x)|\leq L_i|t|$}
The displayed estimate, for all $x$ and real $t$, defines a coordinate <Lipschitz constant> $L_i$ for the <partial derivative> in coordinate $i$. It implies the coordinate <descent lemma>
$$
f(x+te_i)\leq f(x)+t\partial_i f(x)+\frac{L_i}{2}t^2
$$
by the <fundamental theorem of calculus> and integration of the <Lipschitz continuity> estimate. Subtracting an <affine function> leaves these constants unchanged.
The apparently weaker estimate along the singleton negative <gradient> direction is equivalent when every real <step size> is allowed: if the initial <partial derivative> is nonzero, any displacement can be written $-\alpha\partial_i f(x)e_i$; if it is zero but the final <partial derivative> is nonzero, reverse the endpoints; if both vanish the estimate is trivial.
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Target: gradient-descent.bigb, under Lipschitz gradient.
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