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Peano's form: the normalized Taylor remainder tends to zero

Statement

Let $n \geq 1$ . If there is a real $δ > 0$ such that $f$ is $n$ -times differentiable on the open interval $N_{δ} (a) = (a - δ, a + δ)$ , then $\frac{R_{n, a} f (x)}{(x - a)^{n}} ⟶ 0 (x \to a) .$ Equivalently, in the usual little- $o$ shorthand, $f (x) = T_{n, a} f (x) + o ((x - a)^{n})$ . For $n = 0$ , the analogous assertion is the separate continuity condition at $a$ : for every $ε > 0$ , all domain points $x$ sufficiently near $a$ satisfy $∣ f (x) - f (a) ∣ < ε$ .

Facts & Assumptions

Given: The stated differentiability on the open neighbourhood $N_{δ} (a)$ of The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ , or the separate continuity hypothesis when $n = 0$ .

[L1]

Taylor polynomials and their matching derivatives are Taylor polynomials and their remainders and Taylor polynomials match the prescribed derivatives at the centre.

[L2]

The derivative quotient is 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, differentiability implies continuity (A function differentiable at $c$ is continuous at $c$ ), and continuity at $a$ has the stated quantified condition (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point). The Cauchy quotient lemma is Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero, the shifted-power derivative follows from For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term and 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , and finite limits are The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ and obey Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero.

[L3]

For $n \geq 1$ , the canonical real $ι (n)$ is positive and hence nonzero (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · induction

1.1

Under the separate $n = 0$ hypothesis, the quantified assertion is exactly the definition of continuity at $a$ .

givenL2

1.2

For $n = 1$ , the derivative definition gives $\frac{f (x) - f (a) - f^{'} (a) (x - a)}{x - a} = \frac{f (x) - f (a)}{x - a} - f^{'} (a) ⟶ 0.$ This is the required base case.

basegivenL2

1.3

Assume $n \geq 2$ and the assertion through order $n - 1$ . Put $R = R_{n, a} f$ . Then $R (a) = R^{'} (a) = 0$ , and $R^{'} (x) = R_{n - 1, a} (f^{'}) (x)$ .

L1algebra

2.1

By the induction hypothesis applied to $f^{'}$ , $R^{'} (x) / (x - a)^{n - 1} \to 0$ . Applying the Cauchy quotient lemma to $R (x) - R (a)$ and $(x - a)^{n}$ gives $R (x) / (x - a)^{n} = R^{'} (ξ) / (ι (n) (ξ - a)^{n - 1})$ for a point $ξ$ between $a$ and $x$ .

step 1.3ihL2L3

3.1

Since $ξ \to a$ , the right side tends to $0$ . This proves the Peano estimate without assuming continuity of $f^{(n)}$ .

step 2.1L2discharge-induction∎

Depends on

Used by

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Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
Taylor's theorem (Wikipedia): statement and Peano remainder (standard reference, not scraped)