For parametric curves in 2-space, this means that the curve will be continuous if the following condition is met:
For some arbitrarily small εx and εy and for every value t0 of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t0 - &delta ≤ t ≤ t0 + &delta,
x(t0) - εx ≤ x(t) ≤ x(t0) + εx and
y(t0) - εy ≤ y(t) ≤ y(t0) + εy.
A parametric curve in 3-space will be continuous if the following condition is met:
For some arbitrarily small εx, εy, and εz and for every value t0 of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t0 - &delta ≤ t ≤ t0 + &delta,
x(t0) - εx ≤ x(t) ≤ x(t0) + εx,
y(t0) - εy ≤ y(t) ≤ y(t0) + εy, and
z(t0) - εz ≤ z(t) ≤ z(t0) + εz.