Remark 2

Tag 082R, permanent. Cite it as [Bourbaki, Tag 082R].

Functions of a Real Variable, chapter I, § 1. FIRST DERIVATIVE

If a function $\mathbf{f}$ , defined on $I$ , is differentiable at a point $x_0 \in I$ , it is necessarily continuous relative to $I$ at this point.

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

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