Local descent inequality for a Lipschitz gradient on an open segment

Given: Total derivatives and gradients The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, For a differentiable scalar field, $D_{v}f(a)=⟨\mathrm{\nabla }f(a),v⟩$ and the unit direction of steepest ascent is the normalized gradient, the Lipschitz definition and Euclidean metric Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction, ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it, local continuity of totally differentiable maps Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity, Cauchy–Schwarz Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, derivative rules The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, and the mean-value theorem The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$.

The Lipschitz condition is $d_2(\mathrm{\nabla }f(u),\mathrm{\nabla }f(v))\le L{d}_2(u,v)$; for $n\ge 1$, $d_2(a,b)=\mathrm{\parallel }a-b{\mathrm{\parallel }}_2$ (Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction, ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it).

A real function continuous on a closed interval and differentiable in its interior has a point where its derivative equals its secant slope (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$).

Total differentiability implies continuity, and the directional derivative along $d$ is $D_{d}f(a)=⟨\mathrm{\nabla }f(a),d⟩$ (Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity, For a differentiable scalar field, $D_{v}f(a)=⟨\mathrm{\nabla }f(a),v⟩$ and the unit direction of steepest ascent is the normalized gradient).

The identity function has derivative $1$ by its difference quotient; the sum, scalar-multiple, and product rules follow from Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, and restricting a differentiable function to an interval preserves its interior derivatives (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set).

Cauchy–Schwarz and norm homogeneity give $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }sv{\mathrm{\parallel }}_2=\mathrm{\mid }s\mathrm{\mid }\mathrm{\parallel }v{\mathrm{\parallel }}_2$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

Sums, scalar multiples, products, and polynomials of continuous real functions are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).