LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The finite KL generator has the required convex boundary extension

Statement

For $f (t) = t \log t$ on $t > 0$ , one has $f (1) = 0$ , $f$ is convex on $(0, \infty)$ , $\lim_{t ↓ 0} f (t) = 0$ , and $\lim_{t \to \infty} f (t) / t = + \infty$ .

Facts & Assumptions

Given: The function $f : (0, \infty) \to R$ defined by $f (t) = t \log t$ .

[F1]

The natural logarithm as the inverse of the exponential function and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is defined on $(0, \infty)$ , is strictly increasing and onto $R$ , satisfies $\log 1 = 0$ , and $\log (1 / t) = - \log t$ .

[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, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : $\log^{'} (t) = 1 / t$ , and the product rule holds. The identity function has derivative $1$ and constant functions have derivative $0$ , directly from the difference quotient in 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.

[F3]

A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative and Convex and strictly convex functions on Euclidean convex sets: a twice differentiable function on an open interval is convex when its second derivative is nonnegative there.

[F4]

The logarithm grows more slowly than every positive real power: $\log s / s \to 0$ as $s \to + \infty$ .

Proof

1.1F1F2F3givenalgebra

By [F1], $f (1) = \log 1 = 0$ . The difference quotient in [F2] gives derivative $1$ for the identity function and $0$ for a constant function. Applying the product rule and then the logarithm derivative in [F2] yields $f^{'} (t) = \log t + 1$ and $f^{''} (t) = 1 / t > 0$ for every $t > 0$ . Thus [F3] gives convexity on $(0, \infty)$ .

1.2F1F4givenalgebra

Put $s = 1 / t$ . As $t ↓ 0$ , $s \to + \infty$ and [F1] gives $t \log t = - \log (s) / s$ . Its limit is zero by [F4], so $\lim_{t ↓ 0} f (t) = 0$ .

2.1F1givenchoosealgebra∎

For $t > 0$ , $f (t) / t = \log t$ . Since $\log$ is increasing and onto $R$ by [F1], for each real $M$ there exists $a > 0$ with $\log a > M$ , and every $t > a$ has $\log t \geq \log a > M$ . Thus $\lim_{t \to \infty} f (t) / t = + \infty$ .

Depends on

Used by

Finite Kullback–Leibler divergence Definition
Finite data processing for Kullback–Leibler divergence Theorem

Dependency tree · two levels

40 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

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 7, §7.1 (standard reference, not scraped)
Igal Sason and Sergio Verdú, f-Divergence Inequalities, §II-B, equations (43)–(45) (standard reference, not scraped)