ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

How 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.

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The extension of x^2 sin(1/x) by zero is differentiable but its derivative is discontinuous at zero

Example

Define $f (0) = 0$ and $f (x) = x^{2} \sin (1 / x)$ for $x \neq 0$ . Then $f$ is differentiable on $R$ , with $f^{'} (0) = 0$ , but $f^{'}$ is not continuous at $0$ .

Facts & Assumptions

Given: The function $f$ of the statement.

[L1]

$∣ \sin u ∣ \leq 1$ , $\sin^{'} = \cos$ , and the quarter-turn values of sine/cosine hold (Parity and the Pythagorean identity for sine and cosine, The derivatives of sine and cosine are cosine and minus sine, Quarter-turn values and shifts by pi/2 and pi).

[L2]

The chain and derivative-algebra rules, and the derivative definition, hold (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: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[L3]

A function limit implies the same limit along every punctured convergent sequence (Heine criterion: $\lim_{x \to c} f (x) = L$ iff $f (x_{k}) \to L$ for every sequence in $A ∖ {c}$ converging to $c$ ).

Verification

technique · direct

1.1

The difference quotient at zero is $f (x) / x = x \sin (1 / x)$ , whose absolute value is at most $∣ x ∣$ ; hence $f^{'} (0) = 0$ .

L1L2

1.2

For $x \neq 0$ , product and chain rules give $f^{'} (x) = 2 x \sin (1 / x) - \cos (1 / x)$ .

L1L2

2.1

Along $r_{n} = 1 / (2 π (n + 1))$ , the derivative tends to $- 1$ ; along $s_{n} = 1 / ((2 n + 1) π)$ , it tends to $1$ .

step 1.2L1algebra

3.1

Both sequences tend to zero, so [L3] shows that $f^{'}$ has no limit at zero and is not continuous there.

step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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

C. Schmeiser, Introduction to Analysis (standard reference, not scraped)
R. Bartle and D. Sherbert, Introduction to Real Analysis (standard reference, not scraped)
H. Zeisel, lim sin(x)/x and the definition of pi (standard reference, not scraped)