Fractals and the monadic second order theory of one successor #
by PHILIPP HIERONYMI ERIK WALSBERG
Oscillation
Let f : (X, dX) → (Y, dY ) be a function between metric spaces. The oscillation of f at x ∈ X is the supremum of all δ ≥ 0 such that for every ε > 0 there are y,z ∈ X such that dX(x, y) < ε, dX(x,z) < ε and dY (f(y), f(z)) > δ . . Recall that f is continuous at x if and only if the oscillation of f at x is zero.
noncomputable def
oscillation
{X : Type}
{Y : Type}
[MetricSpace X]
[MetricSpace Y]
(f : X → Y)
(x : X)
:
Equations
Instances For
The oscillation of a constant c
is 0 at any x
.
The oscillation of the identity is 0 at any x
.