Continuity
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) Text Given a positive ε we can form the ε
interval about z_{0} = (0,0,f(x(t_{0}),y(t_{0})) on the zaxis, and show
the two planes at levels z_{0} ± ε.
We can then find
a
ρ such that the graph of f(x,y) over the disc of radius ρ
centered at (x(t_{0}),y(t_{0}),0) will lie between the two horizontal planes.
Finally, we can find a &delta so small that if  t  t_{0}  < &delta,
then the image of the parametrics curve (x(t),y(t)) will lie inside the
disc of radius ρ and the curve (0,0,f(x(t),y(t))) will lie between
the
two horizontal planes.
Demos
Continuity of Parametric Curves
 
The value of t_{0} is chosen on a slider bar, as well as
the position of the &delta interval. The value of ρ is chosen so
that the portion of the graph above the disc is between the planes, and
then &delta is chosen so that the image of the parametric curve lies in
the disc of radius ρ.

