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 $c$
• The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

Used by

• Every rational approximation to a real exponent gives the same limiting real power Corollary
• Interior gradient bound for Poisson solutions Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• Polar form of the metric in normal coordinates Corollary
• Polynomial and exponential minimax estimation rates Corollary
• Weak differentiation has a closed graph on its natural domains Corollary
• A power weight fails at both Aₚ endpoints Counterexample
• Boundary-scale derivative blowup despite bounded ball data Counterexample
• Exterior Dirichlet uniqueness needs a far-field condition Counterexample
• Finite speed of propagation does not imply strong Huygens Counterexample
• Outward dilation defeats subcritical inclusion and homogeneous Poincare on Euclidean space Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Subcritical W^1,p is not closed under multiplication Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• Chebyshev weighted spectral Sobolev scale and projection Definition
• Legendre weighted spectral-Sobolev scale and projection Definition
• Quasisymmetric homeomorphisms of the line and circle Definition
• The standard intertwining operator A(nu) Definition
• A 1 over r singularity on the reference triangle Example
• A boundary score cone differs from its tangent space Example
• A Caputo startup singularity and a graded-grid calculation Example
• A one-parameter density prior meets the logarithmic contraction conditions Example
• A regular Bernoulli submodel and its score Example
• A strictly plurisubharmonic exhaustion of the convex unit ball Example
• Cavalieri computes the area of the unit disc from its sections Example
• Direct midpoint quadrature loses order on a half-order singularity Example
• Exponential singular decay, logarithmic balls, and spectral gaps Example
• Finite mean does not give a finite-variance Monte Carlo error Example
• Flux normalization on every centered sphere Example
• Full-history cost and the bias from truncating memory Example
• Gauss–Jacobi regularization for x to the minus one half times log x Example
• Half-space Poisson extension of a plane wave Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newton shell theorem from harmonic mean values Example
• Newtonian potential of radial compact data Example
• Poincare on an interval: the length dependence is linear Example
• Polynomial singular decay with power and logarithmic source balls Example
• Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• Radial powers approach the Morrey borderline exponent Example

…and 92 more results.

Dependency tree · two levels

32 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)