Proposition 3

Tag 084E, permanent. Cite it as [Bourbaki, Tag 084E].

Functions of a Real Variable, chapter I, § 2. THE MEAN VALUE THEOREM

Let $f$ be a vector function with values in a normed space E over $\mathbf{R}$ , defined and continuous on an interval $I \subset \mathbf{R}$ , and right differentiable on the complement B (relative to I) of a countable subset of I; then for all points $x_0 \in B, x \in I, y \in I$ , one has (supposing that $x < y$ , for example)

\| f(y) - f(x) - f'_d(x_0)(y - x) \| \leq (y - x) \sup_{z \in B, x < z < y} \| f'_d(z) - f'_d(x_0) \|.

(9)

Indeed it suffices to apply th. 2, replacing $f$ by the function

f(z) - f'_d(x_0)z,

and $g$ by the linear function whose derivative is $\sup_{z \in B, x < z < y} \| f'_d(z) - f'_d(x_0) \|$ .

Theorem 2 extends to vector functions of a complex variable:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Theorem 2