Alphabeta MathHow 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.
A differentiable function whose derivative is discontinuous
Example
Let be the bounded continuous periodic oscillator of A bounded periodic oscillator made from a quartic Hermite spline, and define , for . Then is differentiable everywhere, but is discontinuous at .
Facts & Assumptions
Given: as displayed.
Derivative algebra, the chain rule, and power derivatives are Sums, scalar multiples, products and quotients: , , , and when , The chain rule, in one line from Carathéodory: if is differentiable at and is differentiable at , then is differentiable at with , and For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term.
Every derivative has the Darboux property (Darboux's theorem: every derivative has the intermediate-value property).
Verification
Since is bounded, , so by The derivative of at a point that is a limit point of , and differentiability on a set.
For , . The periodic piecewise-polynomial derivative takes two separated values along sequences tending to infinity, so has no limit at .
Thus is differentiable and is discontinuous at ; [L2] also confirms that its oscillation is not a jump.
Depends on
- A bounded $C^1$ periodic oscillator made from a quartic Hermite spline
- 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 $g(c) \ne 0$
- 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 $(f \circ g)'(c) = f'(g(c))\,g'(c)$
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Darboux's theorem: every derivative has the intermediate-value property
- 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
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
31 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
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)