nLab modulus of continuity

Definition

Given metric spaces $(X, \delta_X)$ and $(Y, \delta_Y)$,

• given an element $x:A$, a local modulus of continuity for a function $f:X \to Y$ at $x:A$ is a monotonic continuous function on the extended non-negative real numbers $\omega_x \colon [0, \infty] \to [0, \infty]$ whose limit at $0$ is equal to $0$, such that for all elements $y:X$, $d_Y(f(x), f(y)) \leq w_x(d_X(x, y))$.

• a global modulus of continuity for a function $f:X \to Y$ is a monotonic continuous function on the extended non-negative real numbers $\omega:[0, \infty] \to [0, \infty]$ whose limit at $0$ is equal to $0$, such that for all elements $x:X$ and $y:X$, $d_Y(f(x), f(y)) \leq w(d_X(x, y))$.

Related concepts

• pointwise continuous function

• uniformly continuous function

 References

• Wikipedia, Modulus of continuity