Step By Step Calculus » 12.2 - Continuity

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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
\displaystyle x(b)=x(a)\quad\displaystyle x(b)=x(a)\quadand\displaystyle \quad y(b)=y(a)\quad (\displaystyle \quad y(b)=y(a)\quad (or \displaystyle \lim_{x\to b^-}x(t)=x(a)\displaystyle \lim_{x\to b^-}x(t)=x(a) and \displaystyle \lim_{t\to b^-}y(t)=y(a)\displaystyle \lim_{t\to b^-}y(t)=y(a) respectively\displaystyle ). \displaystyle ).
There is a version of the Intermediate Value Theorem for parametric curves: “If x=x(t), \; y=y(t)x=x(t), \; y=y(t) is continuous on [t_1, \; t_2],[t_1, \; t_2], then for any y_ky_k between y(t_1)y(t_1) and y(t_2)y(t_2) there exists an x_k\in\{x(t)\; | \; t\in(t_1,\; t_2)\}x_k\in\{x(t)\; | \; t\in(t_1,\; t_2)\} such that (x_k,\; y_k)(x_k,\; y_k) is a point on the parametric curve. Similarly, for any x_kx_k between x(t_1)x(t_1) and x(t_2)x(t_2) there exists an y_k\in\{y(t) \; | \; t\in (t_1, \; t_2)\}y_k\in\{y(t) \; | \; t\in (t_1, \; t_2)\} such that (x_k, \; y_k)(x_k, \; y_k) is a point on the parametric curve.” We call this the Parametric Intermediate Value Theorem.