LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The first divided difference of a C1 function extends continuously

Statement

Let $a < c < b$ and $f \in C^{1} ([a, b])$ . Define $g (x) = (f (x) - f (c)) / (x - c)$ for $x \neq c$ and set $g (c) = f^{'} (c)$ . Then $g$ is continuous on $[a, b]$ .

Facts & Assumptions

Given: Reals $a < c < b$ and $f \in C^{1} ([a, b])$ .

[F1]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $f \in C^{1}$ means $f$ and its first derivative are continuous on the interval, with one-sided endpoint derivatives.

[F2]

The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set: $f^{'} (c) = \lim_{x \to c,, x \neq c} (f (x) - f (c)) / (x - c)$ .

Proof

1.1givenF1algebra

On $[a, b] ∖ {c}$ the numerator and denominator are continuous and the denominator is nonzero, so their quotient $g$ is continuous at every point other than $c$ .

2.1givenF2algebra∎

By the derivative limit in [F2], the quotient defining $g (x)$ tends to the assigned value $f^{'} (c) = g (c)$ as $x \to c$ . Thus $g$ is continuous at $c$ as well, and hence on the whole interval.

Depends on

Used by

Symmetric Cauchy principal value by subtraction Definition
The Cauchy principal-value subtraction identity Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lloyd N. Trefethen and J. A. C. Weideman, The Exponentially Convergent Trapezoidal Rule, SIAM Review 56(3), 2014 (standard reference, not scraped)