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Limits and Continuity.

Definition 2.5   Let be a vector function, defined on the interval I, with values in the 3-dimensional space, and let be a vector. We say that the vector approaches the vector when t approaches t0, if:

We denote:

Example 2.7   Let . Then:

Definition 2.8   The vector function is continuous at t0 if .

Proposition 2.9

1.
The vector function is continuous at t0 if, and only if, each component Fi is continuous at t0.
2.
The vector function is continuous on the open interval I if it is continuous at every point of I.

Example 2.10

1.
The function is continuous on .
2.
The function is discontinuous at every point (with integer k), and continuous at every other point.

As in Calculus I, we can define one-sided continuity and continuity on a closed interval. Exercise: Have a look at the tutorial for Calculus I, and write down the corresponding definitions here.

Next: Derivatives. Up: Curves in the plane Previous: Vector-valued functions.
Noah Dana-Picard
2001-05-30