Step By Step Calculus » 12.2 - Continuity

Synopsis
The limit of a vector functionx=x(t), \; y=y(t)x=x(t), \; y=y(t) exists as t \rightarrow at \rightarrow a when the limit of each component function exists as t \rightarrow at \rightarrow a. When these limits exist, the limit of the vector function is defined as the limit of its components:
\displaystyle \lim_{t\to a} (x(t),\; y(t))=\left(\lim_{t\to a}x(t),\; \lim_{t\to a}y(t)\right).
\displaystyle \lim_{t\to a} (x(t),\; y(t))=\left(\lim_{t\to a}x(t),\; \lim_{t\to a}y(t)\right).
A vector function x=x(t), \; y=y(t)x=x(t), \; y=y(t) is said to be continuous at aa if \displaystyle \lim_{t\to a} (x(t), \; y(t))=(x(a),y(a)).\displaystyle \lim_{t\to a} (x(t), \; y(t))=(x(a),y(a)).
A parametric curve defined for tt in a finite interval [a \;b][a \;b] of [a, b)[a, b) is said to be closed if its initial point and endpoint coincide and it completely encloses an area. In other words, a parametric curve represented by a continuous vector function (x(t),\; y(t)),\; a\leq t\leq b(x(t),\; y(t)),\; a\leq t\leq b is closed if