Continuity
(Page: 1
 2  3
 4
) Text If the coordinate functions x(t) and y(t) are continuous
functions of the parameter t, then the function that sends t to the
point (x(t),y(t)) is continuous.
This means that for any t_{0} in the domain,
and any positive ε, there is a &delta such that (x(t),y(t)) is
within the disc of radius ε about (x(t_{0}),y(t_{0})) whenever  t  t_{0}  is less than &delta. We achieve this by choosing &delta so
small that  x(t)  x(t_{0})  < ε/2 and  y(t)  y(t_{0}) 
< ε/2, by virtue of the continuity of x(t) and y(t) at t_{0}.
Then √((x(t)x(t_{0}))^{2} + (y(t)y(t_{0}))^{2}) < √(ε^{2}/4 + ε^{2}/4)) = ε/2 < ε
if  t  t_{0}  < &delta.
Demos
Continuity of Parametric Functions
 
Define an interval for t, and two functions x(t), y(t).
In a threedimensional graph, show (t, x(t),0), (t, 0, y(t)), and
(0,x(t),y(t)). For a given t_{0} and interval on the taxis determined
by
a &delta, show the strip above this interval in the first plane and in
the second plane, and show the ε/2 strip about x(t_{0}) in the third plane and the ε/2 strip about y(t_{0} in the third plane, intersecting in a square region completely contained in ε disc about (x{t_{0}),y(t_{0})) in the third plane. Choosing &delta small enough will make the &delta strips lie in the respective ε/2 intervals, so the image of the parametric curve will lie in the square
therefore in the disc.

