Abstract In this paper we
derive a variant of the Lagrange's formula for the vectorvalued
functions of severable variables, which has the form of equality.
Then, we apply this formula to some subtle places in the proof of
the inverse function theorem. Namely, for a continuously
differentiable function $f$, when $f'(a)$ is invertible, the points
$a$ and $b = f(a)$ have open neighborhoods in the form of balls
of fixed radii such that $f$, when restricted to these
neighborhoods, is a bijection whose inverse is also continuously
differentiable. To know the radii of these balls seems to be
something hidden and tricky, but in the proof that we suggest the
existence of such neighborhoods is ensured by the continuity of
the involved correspondences.
