Entropy endpoint limits and the two monotone branches

Given: The variables and hypotheses of the Statement.

Logarithmic tangent bound and a quadratic entropy envelope gives $\log y\le y-1$ for $y>0$.

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm gives continuity, range, strict monotonicity, the product and reciprocal laws, and $\log 1=0$.

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t gives $(\log a)\mathrm{\prime }=1\mathrm{/}a$ for $a>0$.

On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed gives strict monotonicity on an interval for a continuous function whose derivative has the corresponding strict sign at every interior point.

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b) attains every value between the endpoint values of a continuous function on a closed real interval.

Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(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) ≠ 0 gives the sum, scalar-multiple and product rules for functions differentiable on a common domain at a limit point of that domain.

For a natural n ≥ 1 the function x ↦ xⁿ is differentiable everywhere with derivative ι(n) x^ n-1; for n = 0 it is the constant 1, with derivative 0; for a natural n ≥ 1 the function x ↦ x⁻ⁿ is differentiable at every x ≠ 0 with derivative -ι(n) x⁻ⁿ⁻¹; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, with exponents $0$ and $1$, gives the derivatives $0$ and $1$ of the constant function $1$ and the identity function, respectively.

A function differentiable at c is continuous at c gives continuity at every limit point of the domain where the function is differentiable.

Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0} gives, for each $a>0$ in $\mathbb{R}$, the unique nonnegative $s=\sqrt{a}$ with $s^2=a$; necessarily $s>0$.

The left and right limits of f at c, as limits of the restrictions of f to A ∩ (-∞, c) and A ∩ (c, ∞) defines the right limit at $0$ by the condition that for every $\epsilon >0$ there is $\delta >0$ such that $0<a<\delta$ implies $\mathrm{\mid }I(a)-1\mathrm{\mid }<\epsilon$. The point $0$ is a limit point of $(0,\mathrm{\infty })$.

Limits at +∞ and -∞, and infinite limits at a point defines divergence to $+\mathrm{\infty }$ at $+\mathrm{\infty }$ by requiring, for every real $B$, a real $M$ such that $a>M$ implies $I(a)>B$. The domain $(0,\mathrm{\infty })$ is unbounded above.

The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set defines the derivative at a domain limit point by the limit of the difference quotient, and differentiability on a set pointwise. Here $I\mathrm{\prime }\mathrm{\prime }$ denotes the derivative of $I\mathrm{\prime }$.

Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of ℝ, with the dictionary to monotone sequences defines strict increase on a domain by $x<y\Rightarrow f(x)<f(y)$ and strict decrease by $x<y\Rightarrow f(x)>f(y)$; either condition makes the function injective.

Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point defines continuity at $c$ by requiring, for every $\epsilon >0$, a $\delta >0$ such that domain points $x$ with $\mathrm{\mid }x-c\mathrm{\mid }<\delta$ satisfy $\mathrm{\mid }f(x)-f(c)\mathrm{\mid }<\epsilon$. Continuity on a domain is pointwise continuity, and passes to restrictions of that domain.