Higher derivatives and the classes $C^k$ and $C^{\mathrm{\infty }}$

Depends on

• The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
• Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
• The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
• The natural numbers $\mathbb{N}$ (von Neumann)

Used by

• Certified adaptive Simpson from derivative enclosures Algorithm
• Finite endpoint power substitution followed by Gauss–Jacobi quadrature Algorithm
• A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there Corollary
• A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative Corollary
• A uniform limit of smooth functions need not be differentiable anywhere Corollary
• Even-power expansion of trapezoid levels with a uniform remainder Corollary
• A boundary mode has a different concentration rate Counterexample
• A flat smooth real function has no holomorphic extension near zero Counterexample
• A reflected sphere embedding is regularly homotopic but not isotopic to the standard one Counterexample
• A singular interior mode need not have Gaussian scaling Counterexample
• A smooth function not equal to its Maclaurin series Counterexample
• Two equal modes contribute two Gaussian terms Counterexample
• Uniform meshes refute the Cᵈ⁺¹-seminorm-only Floater–Hormann rate Counterexample
• Legendre polynomials by Rodrigues' formula and their unweighted pairing Definition
• Score and Fisher information Definition
• Symmetric Cauchy principal value by subtraction Definition
• Taylor and Maclaurin series Definition
• Taylor polynomials and their remainders Definition
• A bounded C¹ periodic oscillator made from a quartic Hermite spline Example
• A Caputo startup singularity and a graded-grid calculation Example
• A cubic gives equality in the pointwise interpolation remainder Example
• An interior Cauchy principal value reduces to a regular integral Example
• Cancellation of the leading Simpson error coefficient Example
• Fixed-time error slopes for Euler, midpoint, and classical RK4 Example
• For every k≥0, xᵏ|x| is Cᵏ but not Cᵏ⁺¹ Example
• The one-panel trapezoidal bound is attained by x squared Example
• The real function 1/(1+x²) is smooth on the real line but its Maclaurin series has radius one Example
• The square-root endpoint does not have Romberg's even-power order Example
• x↦ x³ changes from concave to convex at zero and has an inflection point there Example
• A Gaussian collocation defect has a high-order weighted endpoint bound Lemma
• A sparse Hamming packing for every 1≤ k≤ d/2 Lemma
• A sufficiently high power regularizes finite logarithmic endpoint factors Lemma
• An ambient isotopy preserves the orientation of an invariant round sphere Lemma
• Bounded variation controls adjacent Hermite–Genocchi windows Lemma
• Chebyshev bounds and endpoint derivatives used by stabilized Runge–Kutta methods Lemma
• Collocation polynomial derivatives scale with the step size Lemma
• Hermite–Genocchi simplex formula for divided differences Lemma
• Higher-order Rolle theorem Lemma
• Leading composite Simpson error coefficient Lemma
• Local CAT(1) of the l² product from a model S²× S² sine-comparison calculation Lemma

…and 34 more results.

Dependency tree · two levels

18 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 I, Taylor's theorem and related calculus (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)