Continuity and derivatives of positive-base real powers

Depends on

• Real powers for positive bases, with the zero-base positive-exponent convention
• The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
• The exponential function is smooth and $(\exp)'=\exp$
• The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$
• Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
• A function differentiable at $c$ is continuous at $c$

Used by

• Every rational approximation to a real exponent gives the same limiting real power Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• Cavalieri computes the area of the unit disc from its sections Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• Strong law estimator of an integrable mean Example
• The arc length of one sine period is 4√2 E(1/√2) Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
• The scaled Gaussian integral and its parameter derivative Example
• The surface generated by rotating y=sin x on [0,π] has area 2π(√2+arsinh1) Example
• Tightness from a uniform moment bound Example
• x⁻ᵃ on (0,1) and on (1,∞) calibrates Lᵖ membership Example
• Absolute real powers are Borel measurable and convex Lemma
• Clarkson inequalities in both exponent ranges Lemma
• The sharp interval bound for Cantor measure Lemma
• A tau-critical graph has no wide pure blockade with cograph pattern Theorem
• An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
• Euler's Beta integral converges exactly for two positive parameters Theorem
• Euler's Gamma integral converges exactly for positive real parameters Theorem
• Finite variance logarithmic rate for iid sums Theorem
• For 0 < p < infinity, the layer-cake formula computes the integral of |f|ᵖ from the distribution function Theorem
• Hadamard three-lines theorem Theorem
• Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains Theorem
• Standard Maclaurin expansions Theorem
• The p-series for a real exponent p converges exactly when p is greater than one Theorem
• The rational-supremum construction of real powers agrees with the exponential construction Theorem
• The real Gamma function is smooth and its derivatives are logarithmic moments Theorem

Dependency tree · two levels

28 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

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)